» Required Reading » Table of Contents » Chapter 3 3.4 Matrix Multiplication
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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 ith row and jth 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 ith row of A and the jth column of B is described as (AB)_ij. See also Section 32.2

Exercises:

3.7 Find the product of the two matrices above.

3.8 Create a spreadsheet that produces the product of any two 2 by 2 matrices when their input is given.

3.9 Do the same for 3 by 3 matrices.

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.

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?