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PROFESSOR STRANG: So today we
move to a topic I really like.

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It's the beginning of
the applications.

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So the particular application
that comes first will be

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springs and masses, a
pretty classical problem.

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But what we're looking for is
how do we model it, what's the

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main framework to look at a
whole series of problems.

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So this number one in
the series and it's the

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most straightforward.
 

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Let me draw it with four
springs connecting

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three masses.
 

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And let me fix both ends.
 

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So this will be a
fixed-fixed picture.

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So the masses have some weight.
 

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The weight pulls
the springs down.

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When there was no weight acting
they were not stretched.

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The masses will
stretch the springs.

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And the question is how much
do those, we're looking

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for the displacements.
 

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How much does mass one go down?
mass two, mass three, and of

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course, essentially the
displacement here is

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zero and here is zero.
 

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I don't know if you can imagine
these masses have acted so that

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the position before gravity was
turned on was somewhere up

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here and then it came here.
 

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So this moved down
by a distance u_1.

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Let's use u for the
displacements.

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So if I look at this main
picture here I have

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displacements, movements,
u_1, u_2, u_3.

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Now what happens physically?
 

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Important in every one of
these examples to see what's

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happening physically.
 

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Of course, this one moved
down by some u_2, this one

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moved down from its total
rest position to u_3.

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These are not oscillating.
 

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Next week they'll start
moving, time will enter.

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Here I'm just looking
for a steady state.

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They come to rest,
they stretch.

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So what's your feeling of
what's going to happen here

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somehow The displacements
look to me like they'll

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all be positive.
 

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What's the key
equation going to be?

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That when this moves down it
will stretch that spring.

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Hooke's Law will say there's a
force, the spring pulls back.

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The spring pulls back with
a force proportional

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to the stretch.
 

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So u_1, u_2, and
u_3 are movements.

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Here is a key question.
 

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What's the stretching
and spring number two?

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So this is spring one,
two, three and four.

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How much does spring number
two stretch? u_2-u_1.

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A difference is
coming in there.

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So let me put that up here.
 

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So stretching or elongation,
I'll use two words, elongation

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I'll say sometimes because
that starts with a letter e.

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So these are the elongations
in the springs, in

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the four springs.
 

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It's the amount the
spring stretches.

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Or what's the opposite
of stretching?

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Compression somehow.
 

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Looks to me like this last
spring, at least, is going to

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be compressed, and I'm not
sure about the others.

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So we've got four springs.
 

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And each one has a stretching
or compression, an elongation.

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And then there's a link then
that you already told me.

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That e_2 is, just from the
picture, e_2 is the difference

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between u_2 and u_1.
 

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Because the lower mass goes
down by distance u_2, the

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upper mass by u_1 and
spring is stretched by

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the difference u_2-u_1.
 

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So that's a first key fact.
 

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So that expresses somehow
a fact of geometry.

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Of sort of the way
things are connected.

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The material properties of
the springs have not got

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into the picture yet.
 

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But now Hooke's Law brings
them into the picture.

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By stretching a spring
that produces a force

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that pulls back.
 

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So we get, can I say,
forces in the spring.

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And let me give those a name
w. w_1, two, three and four.

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And then the link between the
stretching and the force that

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it produces is, so that's
somehow where the properties

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of the material come in.
 

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So I have to say, what are the
properties of the springs?

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So this will be Hooke's
Law, this step.

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Hooke's Law for this
particular application.

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And so I have to say these
springs have spring constants.

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So I haven't completed the
description of the problem

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until I've told you about
springs themselves

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and the masses.
 

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So the spring constants will
be c_1, c_2, c_3, and c_4.

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00:07:00 --> 00:07:03
 
 

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And now what does
Hooke's Law say?

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Usually this physical law in
the middle we keep it linear.

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Of course, we all understand
that it if these springs were

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enormously stretched the
elastic property could

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become non-linear.
 

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It could become plastic.
 

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The first law always
has somebody's name.

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Was the person to see that
in some range of small

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displacements, so I guess
that's the answer.

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We're speaking here about small
displacements, small stretching

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00:07:45 --> 00:07:50
up to the point where Hooke's
Law continues to hold.

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00:07:50 --> 00:07:53
And now what does
Hooke's Law say?

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It says that each force in the
spring is proportional to the

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00:08:03 --> 00:08:07
stretching of the spring.
 

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00:08:07 --> 00:08:11
You could say it's a diagonal
matrix is showing up here.

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The vector of w's, the vector
of forces in the spring is a

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00:08:19 --> 00:08:24
diagonal matrix C, which it
just has these numbers on the

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00:08:24 --> 00:08:32
diagonal, c_4 times the e's.
 

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00:08:32 --> 00:08:38
So of course I'm going to write
that in matrix notation as W

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equals a matrix C times e.
 

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So there in the middle
is the physics.

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The material properties,
the constitutive law.

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C can stand for constants,
for constitutive law,

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later for conductances.
 

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It's the place where
the material enters.

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And now how do we
complete this picture?

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In the end we have to
bring in the masses.

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Gravity is the external force
that's making things happen.

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We need a force term
from outside to move

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us away from zeroes.
 

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And that will be the downward
forces f_1, f_2, f_3

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on the three masses.
 

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So I plan to complete this
picture with a force

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balance equation on the
masses, on each mass.

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When I use the word framework
there, this is what

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00:10:14 --> 00:10:17
I was talking about.
 

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00:10:17 --> 00:10:23
I guess what I want to say is I
really have found that this way

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00:10:23 --> 00:10:33
of describing, modeling the
problem is successful for

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so many applications.
 

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You have somehow a
geometry, a step which'll

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involve a matrix A.
 

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Then you have a physical step
which involves a matrix C.

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And then finally you
have a force balance.

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In a way this force balance or
its analog, the analog would

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00:11:03 --> 00:11:05
be Kirchoff's current law.
 

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00:11:05 --> 00:11:07
We'll see that for networks.
 

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Flow in equals flow out.
 

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00:11:10 --> 00:11:13
Force on one side equals
force on the other.

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If we're talking about
equilibrium we can expect

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our model to have an
equation like that.

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And for me it really helps to
know when a new model comes in.

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Like somebody'll come into my
office with a problem in

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00:11:28 --> 00:11:40
chemistry or biology. but if it
fits in this framework I'll be

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00:11:40 --> 00:11:47
looking for a balance equation,
a continuity equation

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00:11:47 --> 00:11:52
at the end.
 

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This part was easy and
it's these two parts

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that I want to pin down.
 

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Well you told me
how to start here.

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So the elongation, so I want
to take this step again.

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I want to find the elongations
from some matrix that

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multiplies the displacements.
 

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So I'm just completing
this step.

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And you told me what is the
stretching in spring two.

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Again, do you mind
just saying it again?

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The stretching in that second
spring, the amount, it's made

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longer by the action of
gravity was? u_2-u_1.

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00:12:45 --> 00:12:45
u_2-u_1.
 

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So e_2 will be a minus one here
for u_1, a plus one and a zero.

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That will be a typical row of
this matrix, the displacement

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00:13:04 --> 00:13:06
stretching matrix,
you could say.

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Now what about the
stretching in e_1?

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00:13:08 --> 00:13:10
What's the stretching in e_1?
 

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00:13:10 --> 00:13:15
 
 

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00:13:15 --> 00:13:15
Only u_1.
 

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00:13:16 --> 00:13:21
Because essentially it's
u_1-u_0 but u_0 is set

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to zero by the support.
 

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So we only have u_1.
 

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Because that multiplication
just gives us.

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So e_1 is u_1.
 

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00:13:32 --> 00:13:33
e_2 is u_2-u_1.
 

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00:13:35 --> 00:13:37
e_3 is what?
 

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The stretching in
the third spring.

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00:13:43 --> 00:13:45
What is it? u_3-u_2.
 

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So I need a one for u_3
and a minus one for u_2.

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And the stretching in
the fourth spring?

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What's the stretching
in the fourth spring?

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I've sort of, and you have too,
mentally given a plus sign when

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the spring is extended and
a minus sign when

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it's compressed.
 

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Plus for retention,
minus for compression.

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00:14:18 --> 00:14:23
So since I fixed that one,
u_4 was zero, so what do

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00:14:23 --> 00:14:25
I have in this last row?
 

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Just minus u_3.
 

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00:14:26 --> 00:14:31
 
 

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I guess what I'm saying here is
that if we get a systematic

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approach to problems then we
know we're looking for a

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00:14:44 --> 00:14:46
matrix that connects these.
 

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We're looking for the material
constitutive law that does this

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00:14:50 --> 00:14:51
and now we're looking
for this one.

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We kind of know where we are.
 

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00:14:53 --> 00:14:55
What to look for.
 

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00:14:55 --> 00:14:59
And so this matrix is the
matrix I'm going to call A.

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00:14:59 --> 00:15:01
So this is e=Au.
 

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00:15:01 --> 00:15:10
 
 

205
00:15:10 --> 00:15:12
Well one more step to go.
 

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And that will be the
force balance step.

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00:15:16 --> 00:15:22
So now, what's the
equation for balance?

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00:15:22 --> 00:15:25
The external forces
are the masses.

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00:15:25 --> 00:15:31
Well, I guess to get the units
right, it should be mass times

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00:15:31 --> 00:15:34
g, the gravitational constant.
 

211
00:15:34 --> 00:15:38
So let me put external
forces f_1, f_2, and f_3.

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00:15:38 --> 00:15:43
 
 

213
00:15:43 --> 00:15:46
The three masses will be
m_1*g, m_2*g, and m_3*g.

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00:15:46 --> 00:15:51
 
 

215
00:15:51 --> 00:15:55
So those are the
forces from outside.

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00:15:55 --> 00:15:58
Now it's the balance
equation I'm after.

217
00:15:58 --> 00:16:01
So this is in this position.
 

218
00:16:01 --> 00:16:03
It's in equilibrium.
 

219
00:16:03 --> 00:16:05
And what does that tell us?
 

220
00:16:05 --> 00:16:09
That tells us that the total
force on this mass, so I'm

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00:16:09 --> 00:16:14
going to take each mass,
it's like a free body

222
00:16:14 --> 00:16:15
force diagram here.
 

223
00:16:15 --> 00:16:17
I'm looking now at that mass.
 

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I'm saying what forces
are acting on it and I'm

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00:16:21 --> 00:16:23
making them balance.
 

226
00:16:23 --> 00:16:25
So what equation
will that give me?

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00:16:25 --> 00:16:27
So let me write that.
 

228
00:16:27 --> 00:16:30
This is now the force
balance equation.

229
00:16:30 --> 00:16:37
Force balance at each mass.
 

230
00:16:37 --> 00:16:41
How much force is pulling up?
 

231
00:16:41 --> 00:16:44
What's the force pulling up on?
 

232
00:16:44 --> 00:16:47
So this spring is
pulling upwards.

233
00:16:47 --> 00:16:51
And it's pulling
upwards by w_1, right?

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Just getting these
letters right.

235
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The w's were the internal
resisting force, reacting

236
00:17:00 --> 00:17:03
force in the spring.
w_1 is pulling up.

237
00:17:03 --> 00:17:09
What other forces are acting?
w_2 is pulling down.

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00:17:09 --> 00:17:14
And also pulling down is?
 

239
00:17:14 --> 00:17:15
Gravity m_1*g.
 

240
00:17:16 --> 00:17:22
So the balance of forces there
says that w_1, the force up is

241
00:17:22 --> 00:17:25
w_2, the force down and m_1*g.
 

242
00:17:27 --> 00:17:33
And similarly the next one will
have, the next mass if I look

243
00:17:33 --> 00:17:38
just at that I see a force up,
a force down and gravity down.

244
00:17:38 --> 00:17:43
So w_2 will be, well, that's
the pull up will be w_3+m_2*g.

245
00:17:45 --> 00:17:50
And the third one, the force up
on the third one will be the

246
00:17:50 --> 00:17:55
force down on the third one.
so I think those are the

247
00:17:55 --> 00:18:00
equations of force balance
written one at a time.

248
00:18:00 --> 00:18:08
And now, of course I'm
going to write that.

249
00:18:08 --> 00:18:14
So that's three equations
with four w's.

250
00:18:14 --> 00:18:20
So I want to write that as,
I want to bring the W's

251
00:18:20 --> 00:18:21
all to the left-hand side.
 

252
00:18:21 --> 00:18:25
Can I do that?
 

253
00:18:25 --> 00:18:29
Can I just bring those
over with minus signs?

254
00:18:29 --> 00:18:34
And make these equal signs.
 

255
00:18:34 --> 00:18:41
So now we've got internal force
balancing external force.

256
00:18:41 --> 00:18:46
This vector of external forces
is the f's and this is

257
00:18:46 --> 00:18:47
the internal forces.
 

258
00:18:47 --> 00:18:51
Now somewhere there we're
going to see a matrix.

259
00:18:51 --> 00:18:54
So I'm going to write this
equation as some matrix.

260
00:18:54 --> 00:18:57
Well, let's figure out
what that matrix is.

261
00:18:57 --> 00:19:01
So it's shape is what?
 

262
00:19:01 --> 00:19:05
I've got three equations, so I
need three rows in the matrix.

263
00:19:05 --> 00:19:09
I've got four w's so
I need four columns.

264
00:19:09 --> 00:19:16
So it's going to multiply w_1,
w_2, w_3, w_4 to give these

265
00:19:16 --> 00:19:22
three masses, can I call them
f_1, f_2, f_3 just to

266
00:19:22 --> 00:19:26
have a good letter.
 

267
00:19:26 --> 00:19:29
We're almost there.
 

268
00:19:29 --> 00:19:30
What's the matrix?
 

269
00:19:30 --> 00:19:36
What's the matrix for this
final step, the force

270
00:19:36 --> 00:19:38
balance equation?
 

271
00:19:38 --> 00:19:40
I just read it off. w_1-w_2.
 

272
00:19:42 --> 00:19:46
I think I've got that. w_2-w_3.
 

273
00:19:46 --> 00:19:49
 
 

274
00:19:49 --> 00:19:52
Tell me what the second row
of the matrix looks like

275
00:19:52 --> 00:19:53
to give me w_2-w_3.
 

276
00:19:53 --> 00:19:56
 
 

277
00:19:56 --> 00:20:01
0, 1 for the w_2, -1.
 

278
00:20:01 --> 00:20:02
Good.
 

279
00:20:02 --> 00:20:13
And for the third, the
final row? .

280
00:20:13 --> 00:20:17
So that completes
the third piece.

281
00:20:17 --> 00:20:21
If I'd given you the problem as
I did, drawn the problem,

282
00:20:21 --> 00:20:26
described it, you know that
there's going to be a

283
00:20:26 --> 00:20:32
connection between the external
forces and the displacements.

284
00:20:32 --> 00:20:35
But what I'm trying to say
is a good way to see the

285
00:20:35 --> 00:20:39
connection is to see it
in three simple steps.

286
00:20:39 --> 00:20:42
The simple step that gets
you from the displacements

287
00:20:42 --> 00:20:44
to the springs.
 

288
00:20:44 --> 00:20:47
A second step within
the springs.

289
00:20:47 --> 00:20:51
A third step back to the
nodes, you could say,

290
00:20:51 --> 00:20:53
back to the masses.
 

291
00:20:53 --> 00:20:59
And of course, the key question
is, what's that matrix?

292
00:20:59 --> 00:21:02
And do you recognize it?
 

293
00:21:02 --> 00:21:05
Do we need a new name
for that matrix?

294
00:21:05 --> 00:21:08
The matrix in the third step?
 

295
00:21:08 --> 00:21:14
So this third step is going to
be that some matrix times w

296
00:21:14 --> 00:21:19
is f and what's that matrix?
 

297
00:21:19 --> 00:21:22
What's the good name
for us to give it?

298
00:21:22 --> 00:21:25
A transpose is the
best possible name.

299
00:21:25 --> 00:21:31
If we've given this matrix the
name A, the stretching

300
00:21:31 --> 00:21:37
displacement matrix, the strain
in elasticity, this becomes the

301
00:21:37 --> 00:21:40
strains, these become
the stresses.

302
00:21:40 --> 00:21:44
But the beauty is, just
beautiful, that the matrix

303
00:21:44 --> 00:21:49
in this law is the
transpose of this one.

304
00:21:49 --> 00:21:51
So it's A transpose.
 

305
00:21:51 --> 00:21:59
So that's the framework seen
now here for the first time.

306
00:21:59 --> 00:22:04
So the key point was that A and
A transpose both appeared but

307
00:22:04 --> 00:22:08
with physical material
properties, constitutive

308
00:22:08 --> 00:22:10
matrix in between.
 

309
00:22:10 --> 00:22:15
So if we put the pieces
together, then we're golden.

310
00:22:15 --> 00:22:19
And then, let's do an example
to see what actually happened.

311
00:22:19 --> 00:22:31
So the equations were e=Aw,
e=Au, then w=Ce, that's Hooke's

312
00:22:31 --> 00:22:35
Law, and then A transpose-- or
maybe I'll write it

313
00:22:35 --> 00:22:40
as f=A transpose*w.
 

314
00:22:40 --> 00:22:42
 
 

315
00:22:42 --> 00:22:45
That's the three steps.
 

316
00:22:45 --> 00:22:50
So in this problem the source
term showed up at that point.

317
00:22:50 --> 00:22:55
The source term came
from external forces.

318
00:22:55 --> 00:22:57
I've got three equations.
 

319
00:22:57 --> 00:23:00
Now I'm going to put
them together into one.

320
00:23:00 --> 00:23:02
I'll put them into
one equation.

321
00:23:02 --> 00:23:05
So this w I'll just substitute.
 

322
00:23:05 --> 00:23:11
So it's A transpose w
is Ce , and e is Au.

323
00:23:11 --> 00:23:13
So I have A transpose C Au.
 

324
00:23:13 --> 00:23:16
 
 

325
00:23:16 --> 00:23:20
So that's the ultimate.
 

326
00:23:20 --> 00:23:23
That's put the whole
structure together.

327
00:23:23 --> 00:23:27
That's the equation
you have to solve.

328
00:23:27 --> 00:23:35
This would be called
the stiffness matrix.

329
00:23:35 --> 00:23:42
And I use the letter
K for that one.

330
00:23:42 --> 00:23:45
So our equation is Ku=f.
 

331
00:23:46 --> 00:23:56
This is our final equation.
 

332
00:23:56 --> 00:24:03
Well, we didn't know w.
 

333
00:24:03 --> 00:24:05
There are two unknowns here.
 

334
00:24:05 --> 00:24:08
Two physical things
that you want to find.

335
00:24:08 --> 00:24:12
If you're designing a bridge or
a structure you want to know

336
00:24:12 --> 00:24:15
the displacements and then
you want to know the

337
00:24:15 --> 00:24:21
internal forces w.
 

338
00:24:21 --> 00:24:22
It's really beautiful.
 

339
00:24:22 --> 00:24:30
The two unknowns of u and w
are somehow dual, we can

340
00:24:30 --> 00:24:36
work with one, work with
the other, work with both.

341
00:24:36 --> 00:24:40
Oh let me just mention that the
finite element method will fit

342
00:24:40 --> 00:24:46
this framework and somehow this
name stiffness matrix has

343
00:24:46 --> 00:24:51
become famous for finite
elements in structures and

344
00:24:51 --> 00:25:01
then it's just exploded to
appear all over the place.

345
00:25:01 --> 00:25:03
I guess we should look
at A transpose C A.

346
00:25:03 --> 00:25:06
 
 

347
00:25:06 --> 00:25:09
We can see what it looks like.
 

348
00:25:09 --> 00:25:14
And also just from the
way it looks there.

349
00:25:14 --> 00:25:16
So I can write it
out explicitly.

350
00:25:16 --> 00:25:18
I think we want to.
 

351
00:25:18 --> 00:25:21
But at the same time I can
learn something from just

352
00:25:21 --> 00:25:25
seeing how it's put together.
 

353
00:25:25 --> 00:25:27
What can you tell me
about A transpose C A?

354
00:25:28 --> 00:25:30
Let's get the shape first.
 

355
00:25:30 --> 00:25:33
Just to see the shape
of these things.

356
00:25:33 --> 00:25:36
The matrix A is what?
 

357
00:25:36 --> 00:25:39
What's the shape of A?
 

358
00:25:39 --> 00:25:43
It's over here. four by three.
 

359
00:25:43 --> 00:25:45
Four by three.
 

360
00:25:45 --> 00:25:52
And the shape of C was,
three by three is it?

361
00:25:52 --> 00:25:55
Where have I got, that C matrix
better be here somewhere.

362
00:25:55 --> 00:25:58
Oh, no, it's four by four.
 

363
00:25:58 --> 00:25:59
Four springs.
 

364
00:25:59 --> 00:26:02
Of course, it had to be four by
four to do that multiplication.

365
00:26:02 --> 00:26:04
There's the C matrix.
 

366
00:26:04 --> 00:26:06
Four by four, thanks.
 

367
00:26:06 --> 00:26:09
And the A transpose matrix?
 

368
00:26:09 --> 00:26:11
Three by four, thanks.
 

369
00:26:11 --> 00:26:15
So the net result
is three by three.

370
00:26:15 --> 00:26:15
Good.
 

371
00:26:15 --> 00:26:20
So it's a square matrix.
 

372
00:26:20 --> 00:26:22
K is a square matrix.
 

373
00:26:22 --> 00:26:27
What else can you
tell me about it?

374
00:26:27 --> 00:26:30
Now we're going to begin to
use some of the, sort of

375
00:26:30 --> 00:26:34
the matrix preparation.
 

376
00:26:34 --> 00:26:37
These matrices are kind
of friends by now.

377
00:26:37 --> 00:26:42
This is a difference
matrix, somehow.

378
00:26:42 --> 00:26:42
Right?
 

379
00:26:42 --> 00:26:47
The stretchings are differences
and displacements.

380
00:26:47 --> 00:26:49
That's its transpose.
 

381
00:26:49 --> 00:26:53
And then the C matrix, which is
the new thing, sort of the new

382
00:26:53 --> 00:26:59
guy to appear today,
is diagonal.

383
00:26:59 --> 00:27:03
Well if I asked you now,
without writing out the matrix

384
00:27:03 --> 00:27:06
for one more property, it's
square, what else could

385
00:27:06 --> 00:27:08
you tell me about it?
 

386
00:27:08 --> 00:27:12
Symmetric is going to be a very
good guess and let's see why.

387
00:27:12 --> 00:27:15
Why is it symmetric?
 

388
00:27:15 --> 00:27:19
How do we show that that?
 

389
00:27:19 --> 00:27:21
What do I do?
 

390
00:27:21 --> 00:27:25
I take the transpose.
 

391
00:27:25 --> 00:27:32
If I take my K transpose, now
I write it as, what do I do?

392
00:27:32 --> 00:27:34
It's a product of things.
 

393
00:27:34 --> 00:27:39
So when I transpose a product I
have the individual transposes

394
00:27:39 --> 00:27:40
in the opposite order.
 

395
00:27:40 --> 00:27:43
So A, its transpose
comes first.

396
00:27:43 --> 00:27:46
C, its transpose comes next.
 

397
00:27:46 --> 00:27:51
A transpose, its
transpose comes last.

398
00:27:51 --> 00:27:57
So that's just the rules
of matrix transposes.

399
00:27:57 --> 00:27:58
Now what?
 

400
00:27:58 --> 00:28:02
Now I'm ready to use
the wonderful fact of

401
00:28:02 --> 00:28:03
what we've got here.
 

402
00:28:03 --> 00:28:06
So what is C transpose?
 

403
00:28:06 --> 00:28:14
So notice we wanted a symmetric
matrix in the middle to be

404
00:28:14 --> 00:28:16
able to knock that T out.
 

405
00:28:16 --> 00:28:19
And what is A
transpose transpose?

406
00:28:19 --> 00:28:22
That's A.
 

407
00:28:22 --> 00:28:25
We've learned that the thing is
symmetric, that if I transpose

408
00:28:25 --> 00:28:30
it I get it back again.
 

409
00:28:30 --> 00:28:33
We're going to see
more about that.

410
00:28:33 --> 00:28:38
But let me do the
multiplication.

411
00:28:38 --> 00:28:41
So I'm going to take
that, oh, boy.

412
00:28:41 --> 00:28:45
How am I going to do that?
 

413
00:28:45 --> 00:28:48
I want to multiply three
matrices to see what K

414
00:28:48 --> 00:28:52
actually looks like here.
 

415
00:28:52 --> 00:28:55
One question first.
 

416
00:28:55 --> 00:29:01
Eventually the solution, the
short formula for the solution

417
00:29:01 --> 00:29:02
will be u=K inverse f.
 

418
00:29:03 --> 00:29:04
Right?
 

419
00:29:04 --> 00:29:11
So the answer will be u=K
inverse f in matrix notation

420
00:29:11 --> 00:29:16
but I'm looking for numbers.
 

421
00:29:16 --> 00:29:20
And then if I know u then
I know the stretching. e

422
00:29:20 --> 00:29:24
is A times K inverse f.
 

423
00:29:24 --> 00:29:28
And w is, I'm just going
down the list, is C times

424
00:29:28 --> 00:29:30
A times K inverse f.
 

425
00:29:30 --> 00:29:32
We've got everything.
 

426
00:29:32 --> 00:29:34
So that's the key.
 

427
00:29:34 --> 00:29:36
This is the key equation.
 

428
00:29:36 --> 00:29:39
That's the answer.
 

429
00:29:39 --> 00:29:44
Let me ask you about inverses.
 

430
00:29:44 --> 00:29:46
What about K inverse?
 

431
00:29:46 --> 00:29:49
We took three steps.
 

432
00:29:49 --> 00:29:54
Now what if I just ask
you about inverses?

433
00:29:54 --> 00:29:56
This is K inverse that
we would like to know.

434
00:29:56 --> 00:30:04
So again, for inverses I'm
going to start this and

435
00:30:04 --> 00:30:09
I'm going to stop halfway
and you'll tell me why.

436
00:30:09 --> 00:30:13
If you give me a product of
matrices and I don't think

437
00:30:13 --> 00:30:17
particularly much I'll take the
inverse of that times the

438
00:30:17 --> 00:30:25
inverse of that times
the inverse of that.

439
00:30:25 --> 00:30:31
And what's the
matter with that?

440
00:30:31 --> 00:30:35
You would say, why not
just undo each step?

441
00:30:35 --> 00:30:45
Why not find the w's from the
f's and then the e's from the

442
00:30:45 --> 00:30:52
w's by dividing and then
the u's from the e's?

443
00:30:52 --> 00:30:58
Why don't we just go backwards
around the loop rather than

444
00:30:58 --> 00:31:00
what I'm saying we have to do.
 

445
00:31:00 --> 00:31:05
We eventually get this step
across with a matrix K that

446
00:31:05 --> 00:31:13
does all three at once.
 

447
00:31:13 --> 00:31:16
Well sometimes we might be
able to, but I don't think

448
00:31:16 --> 00:31:18
we can in this time.
 

449
00:31:18 --> 00:31:21
What's the trouble with
A that I don't want

450
00:31:21 --> 00:31:26
to write A inverse?
 

451
00:31:26 --> 00:31:28
Well I don't say singular.
 

452
00:31:28 --> 00:31:30
What do I say here?
 

453
00:31:30 --> 00:31:37
Look at this matrix A here.
 

454
00:31:37 --> 00:31:38
It's not square.
 

455
00:31:38 --> 00:31:40
It's not square, that's right.
 

456
00:31:40 --> 00:31:46
So I'm not comfortable, I'm not
willing to write A inverse When

457
00:31:46 --> 00:31:51
A is not a square matrix.
 

458
00:31:51 --> 00:31:55
And this distinction, is the
matrix A square or not,

459
00:31:55 --> 00:31:57
is the first issue.
 

460
00:31:57 --> 00:31:59
It's just the picture.
 

461
00:31:59 --> 00:32:01
Let me show you an example of
where it would be square.

462
00:32:01 --> 00:32:02
May I?
 

463
00:32:02 --> 00:32:05
Before I do this
multiplication, can I jump to

464
00:32:05 --> 00:32:11
a, I'll change the line of
springs in a way

465
00:32:11 --> 00:32:15
that'll change A.
 

466
00:32:15 --> 00:32:16
And let me show
you what happens.

467
00:32:16 --> 00:32:23
Suppose I take out that spring.
 

468
00:32:23 --> 00:32:27
So I've removed the
fourth spring.

469
00:32:27 --> 00:32:31
It's a line of springs now,
hanging from a support.

470
00:32:31 --> 00:32:33
It's a perfectly good problem.
 

471
00:32:33 --> 00:32:37
It's problem two, but it's
a different problem.

472
00:32:37 --> 00:32:39
And what's different now?
 

473
00:32:39 --> 00:32:44
There is no fourth spring.
 

474
00:32:44 --> 00:32:46
If this was my problem,
what would be different?

475
00:32:46 --> 00:32:50
There's no fourth spring.
 

476
00:32:50 --> 00:32:53
So that's gone.
 

477
00:32:53 --> 00:32:56
I just have three springs
stretching from three masses.

478
00:32:56 --> 00:32:59
Then the force
balance is the same.

479
00:32:59 --> 00:33:03
Everything looks the same
except there's no force,

480
00:33:03 --> 00:33:06
there's no fourth spring,
so there's no force

481
00:33:06 --> 00:33:11
there, that's gone.
 

482
00:33:11 --> 00:33:14
And of course, how
does C change?

483
00:33:14 --> 00:33:21
So in my new picture now
I have, let me write

484
00:33:21 --> 00:33:23
now, A transpose C A.
 

485
00:33:25 --> 00:33:29
A is now three by three,
right, I've lost a row.

486
00:33:29 --> 00:33:32
A transpose is now three by
three, I've lost a column,

487
00:33:32 --> 00:33:34
that fourth spring is gone.
 

488
00:33:34 --> 00:33:36
And what is C?
 

489
00:33:36 --> 00:33:43
Well of course there's
no guy here anymore.

490
00:33:43 --> 00:33:50
What I'm trying to say is for
this problem the matrices

491
00:33:50 --> 00:33:54
have become square.
 

492
00:33:54 --> 00:33:56
This would be correct.
 

493
00:33:56 --> 00:34:00
So this is an especially
nice kind of problem.

494
00:34:00 --> 00:34:03
It's called statically
determinate.

495
00:34:03 --> 00:34:08
It means I can determine the
three w's from the three f's.

496
00:34:08 --> 00:34:09
I can go backwards.
 

497
00:34:09 --> 00:34:11
Everything is determined.
 

498
00:34:11 --> 00:34:16
The long word for the
fixed-fixed one, our main

499
00:34:16 --> 00:34:19
example, is statically
indeterminate.

500
00:34:19 --> 00:34:24
I cannot determine four
w's from three forces.

501
00:34:24 --> 00:34:28
I can't determine what these
internal forces are until I

502
00:34:28 --> 00:34:33
put the whole loop
into one matrix K.

503
00:34:33 --> 00:34:37
So that's like a warning,
and at the same time, an

504
00:34:37 --> 00:34:39
important separation.
 

505
00:34:39 --> 00:34:44
A few nice problems where
you don't have too many

506
00:34:44 --> 00:34:47
springs, you don't have
too many bars in a truss.

507
00:34:47 --> 00:34:51
You just have like, the minimum
number to hold it together.

508
00:34:51 --> 00:34:55
Could be statically determinate
and square matrices.

509
00:34:55 --> 00:34:59
But here we're not square.
 

510
00:34:59 --> 00:35:01
Now I go back.
 

511
00:35:01 --> 00:35:05
So that would be fixed-free.
 

512
00:35:05 --> 00:35:05
Right?
 

513
00:35:05 --> 00:35:09
That example that I just
described would be fixed-free

514
00:35:09 --> 00:35:14
and we can kind of carry that
along because we know that what

515
00:35:14 --> 00:35:22
happens is we lose a row and a
column and a c_4 is just not

516
00:35:22 --> 00:35:24
in the picture anymore.
 

517
00:35:24 --> 00:35:29
But now I want to go
back to the fixed-fixed

518
00:35:29 --> 00:35:32
one and finish it.
 

519
00:35:32 --> 00:35:38
So that's got a support
down there, too.

520
00:35:38 --> 00:35:41
Key question, what's
this matrix K?

521
00:35:41 --> 00:35:42
This A transpose C A.
 

522
00:35:42 --> 00:35:46
 
 

523
00:35:46 --> 00:35:49
We know it's a square matrix,
we know it's a symmetric

524
00:35:49 --> 00:35:51
matrix, but it would be
really nice to know

525
00:35:51 --> 00:35:53
what does it look like.
 

526
00:35:53 --> 00:35:55
What does that
matrix look like?

527
00:35:55 --> 00:35:57
Can I do the multiplication?
 

528
00:35:57 --> 00:35:59
So this is going to be K.
 

529
00:35:59 --> 00:36:03
So it starts with
a three by four.

530
00:36:03 --> 00:36:08
1, -1; 1, -1; 1 -1.
 

531
00:36:08 --> 00:36:10
Then its got the four by
four, c_1, c_2, c_3, c_4.

532
00:36:10 --> 00:36:14
 
 

533
00:36:14 --> 00:36:16
And then it's got the transpose
of that, which is the

534
00:36:16 --> 00:36:22
1, -1; 1, -1; 1, -1.
 

535
00:36:22 --> 00:36:32
With zero square, I
didn't write anything.

536
00:36:32 --> 00:36:38
We've got three matrices
to multiply together.

537
00:36:38 --> 00:36:41
What's going to happen here?
 

538
00:36:41 --> 00:36:43
Well, let's see.
 

539
00:36:43 --> 00:36:45
I guess, why don't I
multiply that by that?

540
00:36:45 --> 00:36:47
Can I do that?
 

541
00:36:47 --> 00:36:50
So that's like getting
two steps together.

542
00:36:50 --> 00:36:53
It's going to be easy
because of this.

543
00:36:53 --> 00:36:56
This is usually an easy matrix.
 

544
00:36:56 --> 00:36:57
Often diagonal.
 

545
00:36:57 --> 00:37:00
So when I do that
multiplication, so let me,

546
00:37:00 --> 00:37:05
I'll just copy this guy.
 

547
00:37:05 --> 00:37:13
And now c_1 multiplies that
row, c_2 multiplies this row,

548
00:37:13 --> 00:37:24
c_3 multiplies this row and c_4
multiplies the last row. c_1 in

549
00:37:24 --> 00:37:26
that row, c_2, c_3, and c_4.
 

550
00:37:26 --> 00:37:30
And now I'm ready to put
those together into K.

551
00:37:30 --> 00:37:33
So K will be three by three.
 

552
00:37:33 --> 00:37:34
What does it have?
 

553
00:37:34 --> 00:37:35
It has c_1+c_2.
 

554
00:37:35 --> 00:37:41
 
 

555
00:37:41 --> 00:37:45
And then next to that is going
to be this row one against

556
00:37:45 --> 00:37:50
column two, there'll be a zero
or they'll be a -c_2 here.

557
00:37:50 --> 00:37:55
And then when row one goes
against column three

558
00:37:55 --> 00:38:02
there's nothing.
 

559
00:38:02 --> 00:38:05
Why nothing?
 

560
00:38:05 --> 00:38:10
When do I expect to see a
zero in the overall matrix?

561
00:38:10 --> 00:38:12
What is it about?
 

562
00:38:12 --> 00:38:16
So that zero is in
the position 1, 3.

563
00:38:16 --> 00:38:22
What is it about masses one
and theww that is putting

564
00:38:22 --> 00:38:24
that zero in there.
 

565
00:38:24 --> 00:38:30
We kind of expect to see that
zero even before we find it.

566
00:38:30 --> 00:38:34
If I look at the picture, what
do you notice about masses one

567
00:38:34 --> 00:38:38
and three that is going
to produce the zero?

568
00:38:38 --> 00:38:40
They're not connected.
 

569
00:38:40 --> 00:38:41
They're not connected.
 

570
00:38:41 --> 00:38:45
If I had another spring, which
I could have, connecting mass

571
00:38:45 --> 00:38:50
one to mass three that would
produce, I'd have another.

572
00:38:50 --> 00:38:53
I'd be up to five.
 

573
00:38:53 --> 00:38:55
Instead of four, there'd
be a fifth spring.

574
00:38:55 --> 00:38:57
It would have its own constant.
 

575
00:38:57 --> 00:38:59
It would show up.
 

576
00:38:59 --> 00:39:00
Absolutely could.
 

577
00:39:00 --> 00:39:03
Here we don't have it.
 

578
00:39:03 --> 00:39:04
Now let me keep going.
 

579
00:39:04 --> 00:39:07
I know from symmetry that the
second row times this is

580
00:39:07 --> 00:39:12
going to be zero, is
going to be -c_2.

581
00:39:13 --> 00:39:15
Symmetric as I expected.
 

582
00:39:15 --> 00:39:19
What are you expecting on the
diagonal there? c_2+c_3.

583
00:39:21 --> 00:39:24
That's certainly
the right pattern.

584
00:39:24 --> 00:39:25
Zero, c_2+c_3.
 

585
00:39:27 --> 00:39:28
c_2+c_3.
 

586
00:39:30 --> 00:39:33
And what are you
expecting over here?

587
00:39:34 --> 00:39:36
-c_3 is a good guess.
 

588
00:39:36 --> 00:39:38
It's seeing that pattern.
 

589
00:39:38 --> 00:39:40
Let's just see it happen.
 

590
00:39:40 --> 00:39:47
That second row times this
third guy will give me zero,

591
00:39:47 --> 00:39:51
two rows, two zeroes,
and then a -c_3, good.

592
00:39:51 --> 00:39:56
And now we know the zeroes
going to show up here, the -c_3

593
00:39:56 --> 00:39:57
is going to show up here.
 

594
00:39:57 --> 00:40:01
And what will show
up here? c_3+c_4.

595
00:40:01 --> 00:40:13
 
 

596
00:40:13 --> 00:40:14
So we've got it.
 

597
00:40:14 --> 00:40:18
That's the matrix K that
controls this whole problem.

598
00:40:18 --> 00:40:20
Now we check.
 

599
00:40:20 --> 00:40:22
It's square, yes.
 

600
00:40:22 --> 00:40:25
It's symmetric, yes.
 

601
00:40:25 --> 00:40:31
And notice also it's the kind
of matrix we've seen already.

602
00:40:31 --> 00:40:35
In fact, it's exactly the
matrix we've seen already

603
00:40:35 --> 00:40:38
Suppose all the c's are one.
 

604
00:40:38 --> 00:40:42
Suppose every, c_1,
c_2, c_3, c_4 is one.

605
00:40:42 --> 00:40:51
Then what's the matrix capital
C in that standard case?

606
00:40:51 --> 00:40:56
C will just be the identity
if these are all ones.

607
00:40:56 --> 00:41:00
And then I'm only left
with A transpose A.

608
00:41:00 --> 00:41:08
So let me take that
special case below it.

609
00:41:08 --> 00:41:15
Special IF, so this is IF C is
I, what matrix do we have then?

610
00:41:15 --> 00:41:18
Just to see that we have a
matrix that we know about.

611
00:41:18 --> 00:41:23
So I'm copying this now
here in the case when

612
00:41:23 --> 00:41:24
all the c's are one.
 

613
00:41:24 --> 00:41:28
So if you put all those c's to
be one, what matrix do you get?

614
00:41:28 --> 00:41:30
You get, yes.
 

615
00:41:30 --> 00:41:35
You get the special K.
 

616
00:41:35 --> 00:41:38
Right, you get the special.
 

617
00:41:38 --> 00:41:41
So the work we did to
understand that special

618
00:41:41 --> 00:41:44
matrix pays off here.
 

619
00:41:44 --> 00:41:47
Because we know how
that matrix works.

620
00:41:47 --> 00:41:56
And this matrix, well, it's got
four spring constants in it.

621
00:41:56 --> 00:42:02
But we can guess the
important facts about

622
00:42:02 --> 00:42:04
this one from this one.
 

623
00:42:04 --> 00:42:11
So what are they important
questions about that matrix?

624
00:42:11 --> 00:42:14
This is my matrix K now.
 

625
00:42:14 --> 00:42:17
What would be, we know it's
square, we know it's symmetric.

626
00:42:17 --> 00:42:20
What else do we ask
about a matrix?

627
00:42:20 --> 00:42:25
Well, positive definite, that's
the perfect question, right.

628
00:42:25 --> 00:42:28
And built into positive
definiteness would be a

629
00:42:28 --> 00:42:32
property that we mentioned
the very first day.

630
00:42:32 --> 00:42:34
Is it invertible?
 

631
00:42:34 --> 00:42:36
What's your guess?
 

632
00:42:36 --> 00:42:39
Is that matrix invertible?
 

633
00:42:39 --> 00:42:44
Everybody's going to guess yes
because, you could guess no,

634
00:42:44 --> 00:42:50
where would you be, the
whole course would end.

635
00:42:50 --> 00:42:55
In fact, the world would
end because the problem

636
00:42:55 --> 00:42:57
is correctly posed.
 

637
00:42:57 --> 00:43:00
Those displacements are
determined by the forces

638
00:43:00 --> 00:43:03
and that just says K is
an invertible matrix.

639
00:43:03 --> 00:43:07
So but how do we see that it's
invertible and, even more,

640
00:43:07 --> 00:43:10
positive definite, because
that's the property

641
00:43:10 --> 00:43:11
we now know.
 

642
00:43:11 --> 00:43:14
So why is that matrix
positive definite?

643
00:43:14 --> 00:43:18
Do we want to check
determinants?

644
00:43:18 --> 00:43:21
We could say, ok,
that guy's positive.

645
00:43:21 --> 00:43:29
We could evaluate this product
and find that it came out well.

646
00:43:29 --> 00:43:33
Would you want to do that one?
 

647
00:43:33 --> 00:43:38
We could probably do the
two by two determinant.

648
00:43:38 --> 00:43:42
Could you take that times
that and subtract that?

649
00:43:42 --> 00:43:46
Let's just write it above
what we would get.

650
00:43:46 --> 00:43:50
Just to see it.
 

651
00:43:50 --> 00:43:54
That number times that number
would be a c_1*c_2 and a

652
00:43:54 --> 00:44:02
c_1*c_3 and a c_2*c_2 twice.
 

653
00:44:02 --> 00:44:02
And a c_2*c_3.
 

654
00:44:04 --> 00:44:08
And then I would
subtract off this guy.

655
00:44:08 --> 00:44:13
So it would knock
out that, right?

656
00:44:13 --> 00:44:19
And it would leave something
that would be positive.

657
00:44:19 --> 00:44:21
All this spring constants
are positive here.

658
00:44:21 --> 00:44:27
We're talking normal materials.
 

659
00:44:27 --> 00:44:30
I guess, actually, people
are producing now really

660
00:44:30 --> 00:44:34
amazing materials with
amazing properties.

661
00:44:34 --> 00:44:41
And the amazing property is a
material with a negative c.

662
00:44:41 --> 00:44:47
But that's like 18.085 does
not allow such a thing.

663
00:44:47 --> 00:44:49
Right?
 

664
00:44:49 --> 00:44:52
All these c's are positive.
 

665
00:44:52 --> 00:44:56
And you might guess that the
whole determinant is positive.

666
00:44:56 --> 00:44:58
But now I'd like you
to tell me why.

667
00:44:58 --> 00:45:09
So now we can use our growing
familiarity with matrices to

668
00:45:09 --> 00:45:17
say why is this matrix
positive definite.

669
00:45:17 --> 00:45:21
Is symmetric, of course.
 

670
00:45:21 --> 00:45:27
Positive definite.
 

671
00:45:27 --> 00:45:30
Why?
 

672
00:45:30 --> 00:45:36
So that's what the previous
lecture helped us to answer.

673
00:45:36 --> 00:45:40
We've got these various tests,
but what was the core idea

674
00:45:40 --> 00:45:42
of positive definiteness?
 

675
00:45:42 --> 00:45:46
The core idea was
positive energy.

676
00:45:46 --> 00:45:50
The core idea was I
looked at the energy x

677
00:45:50 --> 00:45:51
trans-- no, u, sorry.
 

678
00:45:51 --> 00:45:56
Have to call it u now.
u transpose times

679
00:45:56 --> 00:46:00
that matrix times u.
 

680
00:46:00 --> 00:46:08
And there was a reason why that
matrix was, why this number,

681
00:46:08 --> 00:46:13
it's going to be
a number, right?

682
00:46:13 --> 00:46:16
This combination will involve
all four of these c's,

683
00:46:16 --> 00:46:19
it'll involve three u's.
 

684
00:46:19 --> 00:46:22
I don't want to write
out that quantity.

685
00:46:22 --> 00:46:26
It would be, I'll have some u_1
squareds and some u_1*u_2.

686
00:46:27 --> 00:46:31
I won't have any u_1*u_3,
because that 1,

687
00:46:31 --> 00:46:33
3 entry is zero.
 

688
00:46:33 --> 00:46:36
But why was this positive?
 

689
00:46:36 --> 00:46:39
Where do I put the parentheses.
 

690
00:46:39 --> 00:46:44
Where do I put the parentheses
to see that that's positive?

691
00:46:44 --> 00:46:50
I put them around where?
 

692
00:46:50 --> 00:46:52
Around that, good.
 

693
00:46:52 --> 00:46:57
And around this?
 

694
00:46:57 --> 00:47:01
This is really, since we now
have a letter for Au, this is

695
00:47:01 --> 00:47:09
really e transpose Ce, right?
 

696
00:47:09 --> 00:47:09
That's e.
 

697
00:47:09 --> 00:47:13
This is eAu and this
is its transpose.

698
00:47:13 --> 00:47:17
And now what?
 

699
00:47:17 --> 00:47:21
So now we've narrowed
it down to C.

700
00:47:21 --> 00:47:26
Oh, we can actually see
why it's an energy.

701
00:47:26 --> 00:47:28
Remember C is that
diagonal matrix.

702
00:47:28 --> 00:47:29
What will this be?
 

703
00:47:29 --> 00:47:37
This is the row of stretchings,
the diagonal matrix of c's and

704
00:47:37 --> 00:47:40
the column of stretchings.
 

705
00:47:40 --> 00:47:45
And now if I do that
multiplication, what do I get?

706
00:47:45 --> 00:47:46
Do you see it?
 

707
00:47:46 --> 00:47:49
Because the physics
is coming in.

708
00:47:49 --> 00:47:53
What do I get?
 

709
00:47:53 --> 00:47:54
This will multiply that.
 

710
00:47:54 --> 00:47:58
So what's the first term I
should write here? e_1?

711
00:47:58 --> 00:48:01
 
 

712
00:48:01 --> 00:48:03
What will it be?
 

713
00:48:03 --> 00:48:04
I only have diagonal.
 

714
00:48:04 --> 00:48:06
In other words, I only
have perfect squares when

715
00:48:06 --> 00:48:08
I look at this thing.
 

716
00:48:08 --> 00:48:16
I think I just have c_1*e_1
squared coming from

717
00:48:16 --> 00:48:17
that diagonal.
 

718
00:48:17 --> 00:48:21
That c_1 there, this e_1 here,
this e_1 there is going

719
00:48:21 --> 00:48:22
to give me that c_1.
 

720
00:48:22 --> 00:48:28
What else will I have? c_2*e_2
squared, c_3*e_3 squared

721
00:48:28 --> 00:48:30
and c_4*e_4 squared.
 

722
00:48:30 --> 00:48:36
 
 

723
00:48:36 --> 00:48:39
And do you remember about
springs and Hooke's

724
00:48:39 --> 00:48:42
Law and energy?
 

725
00:48:42 --> 00:48:44
What's the energy in a spring?
 

726
00:48:44 --> 00:48:48
This is a stretched spring.
 

727
00:48:48 --> 00:48:55
So the energy in a stretched
spring, what I wanted to say,

728
00:48:55 --> 00:48:59
this is the sum of four
internal energies in the four

729
00:48:59 --> 00:49:05
springs but it properly
should have a factor 1/2.

730
00:49:05 --> 00:49:10
There probably, to really use
the word energy properly, it

731
00:49:10 --> 00:49:14
should be 1/2 of all this,1/2
of all this, 1/2 of that's the

732
00:49:14 --> 00:49:18
energy in the first spring, the
energy in the second, the

733
00:49:18 --> 00:49:24
energy in the third and
the energy in the fourth.

734
00:49:24 --> 00:49:30
But of course our matrix
point was, it's positive.

735
00:49:30 --> 00:49:34
It's a sum of squares
multiplied now by these

736
00:49:34 --> 00:49:39
positive numbers, these
elastic constants, c_1,

737
00:49:39 --> 00:49:42
two, three and four.
 

738
00:49:42 --> 00:49:53
So we know the main facts
about that matrix.

739
00:49:53 --> 00:49:58
We're really at the point here
of we've got some problem

740
00:49:58 --> 00:50:03
formulated, we've got the
essential facts about the

741
00:50:03 --> 00:50:07
matrix, it's symmetric,
positive definite,

742
00:50:07 --> 00:50:12
certainly invertible.
 

743
00:50:12 --> 00:50:22
Then there'd be the step of
actually computing U by solving

744
00:50:22 --> 00:50:24
the stiffness equation.
 

745
00:50:24 --> 00:50:30
Say, for example, Professor
Bathe big finite

746
00:50:30 --> 00:50:34
element code, ADINA.
 

747
00:50:34 --> 00:50:38
What's the big picture
for ADINA for any big

748
00:50:38 --> 00:50:39
finite element code?
 

749
00:50:39 --> 00:50:43
NASTRAN, ANSYS, whatever.
 

750
00:50:43 --> 00:50:44
Abacus.
 

751
00:50:44 --> 00:50:46
There are so many
really good ones.

752
00:50:46 --> 00:50:50
And they've taken years and
years of work to create.

753
00:50:50 --> 00:50:56
But if you look to see what are
the elements that go in, you

754
00:50:56 --> 00:51:03
choose the model, and we'll see
in the next chapter, in October

755
00:51:03 --> 00:51:08
we'll see what finite elements
is about, you have the

756
00:51:08 --> 00:51:15
material properties, you
assemble the matrix K.

757
00:51:15 --> 00:51:20
That's a key step, is
assembling this matrix K.

758
00:51:20 --> 00:51:23
And then the final step
is solve the system.

759
00:51:23 --> 00:51:23
Ku=f.
 

760
00:51:24 --> 00:51:28
But it's assembling
that matrix.

761
00:51:28 --> 00:51:30
Now one thing popped
into my head.

762
00:51:30 --> 00:51:34
Do I have time to
mention it or not?

763
00:51:34 --> 00:51:39
And there's no class
Monday I think, right?

764
00:51:39 --> 00:51:41
Can I mention?
 

765
00:51:41 --> 00:51:47
Can you hang on one more second
to mention a really remarkable

766
00:51:47 --> 00:51:51
way to do matrix
multiplication.

767
00:51:51 --> 00:51:54
You may say, we know
matrix multiplication.

768
00:51:54 --> 00:51:55
We got it.
 

769
00:51:55 --> 00:51:55
Right?
 

770
00:51:55 --> 00:51:58
We did it and we got
the right answer.

771
00:51:58 --> 00:51:59
Can I just show
you another way.

772
00:51:59 --> 00:52:03
And you can like,
see if it works.

773
00:52:03 --> 00:52:07
I did this multiplication
by like, I'll say

774
00:52:07 --> 00:52:08
rows times columns.
 

775
00:52:08 --> 00:52:11
I took rows times columns.
 

776
00:52:11 --> 00:52:13
That's the usual way.
 

777
00:52:13 --> 00:52:17
But finite elements and
other, often the right

778
00:52:17 --> 00:52:18
way is the opposite.
 

779
00:52:18 --> 00:52:23
It's columns times rows.
 

780
00:52:23 --> 00:52:26
And of course, this
guy's in here too.

781
00:52:26 --> 00:52:33
You might say, ok, what do I
get from column one times

782
00:52:33 --> 00:52:36
that number, times row one?
 

783
00:52:36 --> 00:52:44
Can you do that multiplication
just mentally?

784
00:52:44 --> 00:52:46
Multiply that column
by that row.

785
00:52:46 --> 00:52:51
First of all, what shape
will the answer have?

786
00:52:51 --> 00:52:54
What shape will the answer
have if I multiply a three

787
00:52:54 --> 00:52:57
by one times a one by three.
 

788
00:52:57 --> 00:52:58
Three by three.
 

789
00:52:58 --> 00:52:59
It's a full matrix.
 

790
00:52:59 --> 00:53:00
Columns times rows.
 

791
00:53:00 --> 00:53:05
And it's a totally legitimate
way to multiply matrices.

792
00:53:05 --> 00:53:09
That column times
that row will be?

793
00:53:09 --> 00:53:11
Well you can see
what will it be.

794
00:53:11 --> 00:53:18
And then the c_1 is
going to come into it.

795
00:53:18 --> 00:53:20
If I just did those
multiplications, it

796
00:53:20 --> 00:53:22
would just be that.
 

797
00:53:22 --> 00:53:28
And then the c_1
puts that there.

798
00:53:28 --> 00:53:30
What do I see there?
 

799
00:53:30 --> 00:53:34
I see the element matrix.
 

800
00:53:34 --> 00:53:37
Do you see that this is
the piece that involved

801
00:53:37 --> 00:53:40
c_1 in the answer?
 

802
00:53:40 --> 00:53:45
Well I guess you'll see it
better when I do column two

803
00:53:45 --> 00:53:48
times c_2 times row two.
 

804
00:53:48 --> 00:53:50
So I have to add that guy on.
 

805
00:53:50 --> 00:53:54
And then I'll leave the other.
 

806
00:53:54 --> 00:53:59
What do I get if I do that
column, three by one, times

807
00:53:59 --> 00:54:02
itself as a row times the c_2.
 

808
00:54:03 --> 00:54:06
I don't know if you see
what I'm going to get.

809
00:54:06 --> 00:54:12
If you just do that, you'll
see a c_2 will appear here.

810
00:54:12 --> 00:54:15
And a -c_2 will appear there.
 

811
00:54:15 --> 00:54:17
And these will be zeroes.
 

812
00:54:17 --> 00:54:20
So this was column
one times row one.

813
00:54:20 --> 00:54:22
This is column two
times row two.

814
00:54:22 --> 00:54:24
And third and then the fourth.
 

815
00:54:24 --> 00:54:29
But do you see that this part
is telling me all about

816
00:54:29 --> 00:54:30
the second spring?
 

817
00:54:30 --> 00:54:33
This part is telling me,
what does the first spring,

818
00:54:33 --> 00:54:36
the c_1, contribute to K.
 

819
00:54:36 --> 00:54:39
This part tells me what
does the c_2 part, do you

820
00:54:39 --> 00:54:41
see the c_2 part in K?
 

821
00:54:41 --> 00:54:44
There, there, minus
there and minus there.

822
00:54:44 --> 00:54:49
The third part from the column
row would be the c_3 part.

823
00:54:49 --> 00:54:51
And the fourth part
from the fourth spring

824
00:54:51 --> 00:54:52
would be the c_4 part.
 

825
00:54:52 --> 00:54:56
So that's a way you
won't have thought of.

826
00:54:56 --> 00:55:00
But it's the way ADINA would
assemble this matrix.

827
00:55:00 --> 00:55:02
It would not do that
multiplication.

828
00:55:02 --> 00:55:05
It would do it this way,
columns times rows.

829
00:55:05 --> 00:55:06
We'll see it again.
 

830
00:55:06 --> 00:55:10
So, hope you have a great
weekend and a holiday Monday

831
00:55:10 --> 00:55:13
that we all happy about.
 

832
00:55:13 --> 00:55:13