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BEN HARRIS: Hi.
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I'm Ben.
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Today we are going to do an
LU decomposition problem.
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Here's the problem right here.
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Find that LU decomposition
of this matrix A.
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Now notice that this matrix
A has variables, as well
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as numbers.
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So the sentence
ends: when it exists.
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And the second part
of the question
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asks you: for which
real numbers a and b
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does the LU decomposition of
this matrix actually exist?
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Now, you can hit pause now and
I'll give you a few seconds.
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You can try to solve
this on your own,
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and then we'll be back
and we can do it together.
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And we're back.
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Now, what do you have to
remember when doing an LU
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decomposition problem?
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Well, we do elimination
in the same way
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that we did before in order to
find U. But with this question
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we need to find L as well.
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So we need to do
elimination, but we also
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need to keep track of
the elimination matrices
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along the way.
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Good.
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So let's do that.
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So let me put my matrix up here.
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And we want to do elimination.
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So which entry do
we eliminate first?
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That's right.
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It's this (2, 1) entry.
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So we replace the second
row by the second row
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minus a times the first
row, and we get this.
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But we're not just
doing elimination,
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we're finding an
LU decomposition.
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So we need to keep
track of the matrix
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that I multiplied the
elimination matrix, that I
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multiplied this matrix A by on
the left to get this matrix.
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So what is that?
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That's this E_(2,1).
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Since I eliminated the (2, 1)
entry, I'll call it E_(2,1).
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And it's this matrix.
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Why is it this matrix?
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Well, remember how
multiplication on the left
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works.
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I replaced the first row
by just the first row.
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I replaced the second
row by the second row
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minus a times the first row.
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So you can just read
off from these rows
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which operations I did.
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Now, which entries
should we eliminate next?
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We need to eliminate this b.
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So we will replace the
third row by the third row
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minus b times the first row.
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And which elimination
matrix did we use?
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Well, note, we
replaced the third row
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by the third row minus
b times the first row.
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That's exactly what you should
read off this elimination
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matrix.
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Good.
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Now, we only have one step left.
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We only need to
eliminate one last entry.
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But this one's a little
tricky, so let's be careful.
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In order to eliminate this
b, we need a to be a pivot.
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In particular, we
need a to be nonzero.
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If a were zero
here, then we would
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have to do a row exchange.
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And that's no good.
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You can't find an
LU decomposition
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if you have to do a row
exchange in elimination.
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So we need to assume
that a is non-zero
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in order to keep going.
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So let's just assume
there that a is non-zero.
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Now, what do we do?
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Well we can replace
the third row
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by the third row minus b
over a times the second row.
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And we just get this.
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a minus b.
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And let's write down
our elimination matrix.
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E_(3,2) now.
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There's our elimination matrix.
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We replaced the third row
by the third row minus b
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over a times the second row.
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Good.
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So we found our U matrix.
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That's what elimination does,
it gives us our U matrix.
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So let me write it up here.
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1, 0, 1; 0, a, 0;
0, 0, a minus b.
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Good.
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Now we have to
find our L matrix,
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and we need to use these
elimination matrices
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that we've been recording along
the way in order to do that.
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So remember that we started
with A, and we got U.
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And how did we do that?
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Well we multiplied on the left
by all of these elimination
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matrices, E_(2,1),
E_(3,1) and E_(3,2).
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Sorry if that's
scrunching together there.
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Now, if we move these
elimination matrices
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to the other side then we'll
get L. So what do we have?
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We have A equals E_(2,1)
inverse, E_(3,1) inverse,
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E_(3,2) inverse times
U. And this is our L,
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this product of
these three matrices.
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Good.
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So let's compute it now.
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So L is the product
of three matrices.
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I need to get them by going
back and looking at these three
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elimination matrices and
taking their inverses.
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Well the nice thing
about taking an inverse
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of an elementary
matrix like this is we
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just make a minus a
plus or a plus a minus.
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So that's easy enough.
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We just change the
off-diagonal entries,
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we just change their signs.
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You can check that that
does what we wanted it to.
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It gives us the inverse.
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Good.
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And the last comment is
that multiplying these three
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matrices is really
easy in this order.
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Turns out all you do is you just
plop these entries right in.
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1, 1.
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Good.
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So this is our L matrix.
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So now we have our U
matrix and our L matrix,
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and we're done with the
first part of the question.
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The second part asks us for
which real numbers a and b
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does this decomposition exist?
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Now let's go back and
remember that at one point
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we had to assume
that A was non-zero.
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That was the only
assumption we had to make
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to get this decomposition.
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So it exists-- it being
this decomposition--
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when a is non-zero.
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And that's the answer
to the second part.
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So we have our LU decomposition,
and we know when it exists.
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Before I end, two comments.
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First, always check your work.
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Always go back and
multiply L times U
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and make sure it's
A, because it's
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easy to screw up the
elimination process
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and it's easy to
check your work.
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So if you go back and make sure
things are as they should be.
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Second comment is that
you might be worried
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when you do this elimination
process that-- well OK, we
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had to assume a is
non-zero because we
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wanted a non-zero pivot.
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You might worry
that we might have
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to have a minus b be non-zero.
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But in fact, a minus b can be 0.
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It's not a problem
for this entry
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to be 0 because we don't have
to do a row exchange to get
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U. That's the only time when we
can't do the LU decomposition.
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In particular, singular matrices
can have LU decompositions.
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Good.
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Thanks.