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_ij.

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

If matrix A has the same number of columns as B has rows, we define the product matrix, AB 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 (AB)_ij. 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 AB?

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 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 |v> ; as a row, <v|. The square of the length of v can then be written as the matrix product <v||v>.

A vector v is an eigenvector of a matrix M when Mv is a multiple of v. The multiple is called the eigenvalue of M having eigenvector v. If the eigenvalue is s, then we have Mv = sv.

The applet here allows you to enter any 2 by 2 matrix, and move the vector v around. When Mv lines up with v, v is an eigenvector of M with real eigenvalue which is given by the ratio of the length of Mv (called v' in the applet) to that of 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?