3.4 Matrix Multiplication

]> 3.4 Matrix Multiplication

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3.4 Matrix Multiplication

A rectangular array of numbers, say $n$ by $m$ , is called a matrix. The i-j-th element of the matrix $A$ is the element in the i-th row and j-th column, and is denoted as $A_{i j}$ .

Here are examples of matrices one two by two and the other two by three

(\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix})

(\begin{array}{l} \begin{matrix} 1 & 4 & 2 \end{matrix} \\ \begin{matrix} 2 & 1 & 0 \end{matrix} \end{array})

If matrix $A$ has the same number of columns as $B$ has rows, we define the product matrix, $A B$ to be the matrix whose elements are dot products between the rows of $A$ and the columns of $B$ . The element obtained by taking the dot product of the i-th row of $A$ and the j-th column of $B$ is described as ${(A B)}_{i j}$ . See also Section 32.2 for a fuller discussion of matrices and their properties.

Exercises:

3.7 Find the product of the two matrices above.

3.8 Build a spreadsheet that multiplies 4 by 4 matrices. Solution

3.9 In exercise 3.8:

1. Where is the matrix product $A B$ ?

2. What appears in columns $p, q, r$ and $s$ in the first four rows?

If you change any of the entries in $A$ or $B$ the product will change automatically, so you have built an 4 by 4 matrix automatic product finder.

3. Can you use this to find the product of a 2 by 3 matrix and a 3 by 4 one? How?

4. Find the tenth power of a matrix $A$ using your product finder. (Hint: use it for $A$ and for $B$ and look in the right place and you have it.)

A vector $\overset{⟶}{v}$ can be written either as a matrix consisting of a single row, or of a single column. When writing it as a column we will write $| \overset{⟶}{v} >$ ; as a row, $< \overset{⟶}{v} |$ . The square of the length of $\overset{⟶}{v}$ can then be written as the matrix product $< \overset{⟶}{v} | | \overset{⟶}{v} >$ .

A vector $\overset{⟶}{v}$ is an eigenvector of a matrix $M$ when $M \overset{⟶}{v}$ is a multiple of $\overset{⟶}{v}$ . The multiple is called the eigenvalue of $M$ having eigenvector $\overset{⟶}{v}$ . If the eigenvalue is $s$ , then we have $M \overset{⟶}{v} = s \overset{⟶}{v}$ .

The applet here allows you to enter any 2 by 2 matrix, and move the vector $\overset{⟶}{v}$ around. When $M \overset{⟶}{v}$ lines up with $\overset{⟶}{v}$ , $\overset{⟶}{v}$ is an eigenvector of $M$ with real eigenvalue which is given by the ratio of the length of $M \overset{⟶}{v}$ (called $\overset{⟶}{v'}$ in the applet) to that of $\overset{⟶}{v}$ , with a sign that is positive when they point in the same direction.

Exercise 3.10 Choose a symmetric matrix and use the applet to determine the two eigenvectors, approximately. Draw them on a piece of paper. Can you notice something about them? What?