Raising the bar

Vector and matrix multiplication

Don't jump to conclusions, and you will be fine. See Matrix notation and summation for the basic notation.

Vector multiplication

Intuition. When I ask students how they would multiply two vectors, by analogy with vector summation they suggest

(1) $latex DC=\left(\begin{array}{ccccc}c_1d_1&...&c_jd_j&...&c_nd_n\end{array}\right).$

This is a viable definition and here is the situation when it can be used. Let $latex d_j$ denote the initial deposit (the principal) of client $latex j$ of a bank. Assuming that clients deposited their money at different times and are paid different interest rates, the bank now owes client $latex j$ the amount $latex c_jd_j$ where the factor $latex c_j$ depends on the interest rate and the length of time the money has been held in the deposit. Thus, (1) describes the amounts owed to customers on their deposits.

However, one might ask how much in total the bank owes on all deposits, and the answer will be given by

(2) $latex D\cdot C=c_1d_1+...+c_jd_j+...+c_nd_n.$

Definition. (2) is called a dot product (because the dot is used as the multiplication sign) or a scalar product (because the result is a scalar). Although we provided a real life situation when (1) is useful, in fact there are deep mathematical reasons for using (2) instead of (1).

Exercise 1. Do you think that $latex A\cdot B=B\cdot A$ for any two vectors $latex A,B$ (symmetry of the scalar product)?

Matrix multiplication

We start with a special case that gave rise to the whole matrix algebra.

Global idea 2. We want to write a linear system of equations

(3) $latex \left\{\begin{array}{c}ax_1+bx_2=e\\cx_1+dx_2=f\end{array}\right.$

in a form $latex Ax=B$.

Put $latex A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$, $latex x=\left(\begin{array}{c}
x_1\\x_2\end{array}\right)$, $latex B=\left(\begin{array}{c}e\\f\end{array}\right).$ Define the product $latex Ax$ by

(4) $latex Ax=\left(\begin{array}{c}ax_1+bx_2\\cx_1+dx_2\end{array}\right).$

Then, taking into account the definition of equality of two matrices, we see that (3) is equivalent to

(5) $latex Ax=B.$

Digesting definition (4). Denote $latex A_1=\left(\begin{array}{cc}a&b\end{array}\right),$ $latex A_2=\left(\begin{array}{cc}c&d\end{array}\right)$ the rows of $latex A.$ Then $latex A$ can be written as $latex A=\left(\begin{array}{c}A_1\\A_2\end{array}\right).$ Such a representation of a large matrix in terms of smaller submatrices is called partitioning. Then (4) shows that elements of the product are dot products of vectors

(6) $latex Ax=\left(\begin{array}{c}A_1\cdot x\\A_2\cdot x\end{array}\right).$

Note that this definition is correct because the number of elements of $latex A_1,$ $latex A_2$ is equal to the number of elements of $latex x.$ Alternatively, the number of columns of $latex A$ equals the number of rows of $latex x.$

General definition

It consists of two parts:

Part 1 (compatibility rule, or rule for counting dimensions) $latex A_{n\times m}B_{m\times k}=C_{n\times k}$ (the number of columns $latex m$ of $latex A$ equals the number of rows $latex m$ of $latex B$).

Part 2 (rows by columns rule, or rule for multiplication) Let us partition $latex A$ into rows and $latex B$ into columns:

$latex A=\left(\begin{array}{c}A_1 \\... \\A_n\end{array}\right),$ $latex B=\left(B^1 \ ... \ B^k\right).$

Then the elements of the product $latex C$ are found as dot products of the rows of $latex A$ by columns of $latex B$:

$latex C=\left(\begin{array}{ccc}A_1\cdot B^1&...&A_1\cdot B^k \\...&...&... \\A_n\cdot B^1&...& A_n\cdot B^k\end{array}\right).$

In words: to find the elements of the first row of $latex C$, fix the first row in $latex A$ and move right along columns of $latex B$.

Cayley appears to be the inventor of this rule, although many other parts of matrix algebra had been discovered before him.

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