Matrix Algebra Whirlwind Tour

Introduction

In this hopefully short and crisp tour, we will look at elementary operations on matrices: transpose, addition/subtraction and multiplication, and finally the inverse.

What is a Matrix?

An rectangular array of numbers, arranged with rows and columns. A few examples are:

Equation 1 is a 3 x 3 matrix (3 rows and 3 columns).

$$\begin{array}{}\text{(1)}& \left[\begin{array}{c}2,3,3\\ 3,1,2\\ 1,2,4\end{array}\right]\end{array}$$

Equation 2 is a 3 x 4 matrix (3 rows and 4 columns).

$$\begin{array}{}\text{(2)}& \left[\begin{array}{c}2,3,3,4\\ 3,1,2,3\\ 1,2,4,1\end{array}\right]\end{array}$$

Equation 3 is a 1 x 4 matrix (1 rows and 4 columns).

$$\begin{array}{}\text{(3)}& \left[\begin{array}{c}2342\end{array}\right]\end{array}$$

Equation 2 is a 3 x 1 matrix (3 rows and 1 columns).

$$\begin{array}{}\text{(4)}& \left[\begin{array}{c}2\\ 3\\ 1\end{array}\right]\end{array}$$

Transpose

A matrix transpose is accomplished by writing rows as columns and vice versa. Hence an $rxc$ matrix becomes transposed into a $cxr$ matrix, as shown below:

c: ⎡a  c  1⎤
   ⎢       ⎥
   ⎣b  d  2⎦

c: ⎡a  b⎤
   ⎢    ⎥
   ⎢c  d⎥
   ⎢    ⎥
   ⎣1  2⎦

Addition and Subtraction

Matrices can be added or subtracted when they are of the same dimensions, rows and columns. Matrix elements are added or subtracted in element-wise fashion.

Addition

c: ⎡a  0⎤
   ⎢    ⎥
   ⎣b  1⎦

c: ⎡a  c⎤
   ⎢    ⎥
   ⎣b  1⎦

c: ⎡2⋅a  c⎤
   ⎢      ⎥
   ⎣2⋅b  2⎦

Subtraction

c: ⎡a  0⎤
   ⎢    ⎥
   ⎣b  1⎦

c: ⎡a  c⎤
   ⎢    ⎥
   ⎣b  1⎦

c: ⎡0  -c⎤
   ⎢     ⎥
   ⎣0  0 ⎦

Multiplication

Matrix multiplication can be visualized in many ways!! According to Gilbert Strang’s famous book on Linear Algebra, there are no less than 4 very useful ways. Let us understand perhaps the easiest one, Linear combination of Columns.

Suppose there are three students who bought two kinds of food and drink:

• Student A: 2 samosas and 1 chai
• Student B: 3 samosas and 0 chai
• Student C: 0 samosa and 2 chai-s (Just before Arvind’s R class.)

Samosas cost 20 and chai costs 10. How much do they pay?

$$\left[\begin{array}{cc}2& 1\\ \\ 3& 0\\ \\ 0& 2\end{array}\right]$$

$$\left[\begin{array}{c}20\\ \\ 10\end{array}\right]$$

$$\left[\begin{array}{c}50\\ \\ 60\\ \\ 20\end{array}\right]$$

$$A$$

$$B$$

$$C$$

We see that the first number $20$ in B multiplies the entire first column in A, and the second number $10$ multiplies the entire second column in A. These two multiplied columns are added to get the matrix C.

Suppose they had two options for shops? And the prices were different?

• Samosas cost 20 and chai costs 10 at Shop#1. (as before)
• Samosas cost 15 and chai costs 15 at Shop#2.

$$\left[\begin{array}{cc}2& 1\\ \\ 3& 0\\ \\ 0& 2\end{array}\right]$$

$$\left[\begin{array}{cc}20& 15\\ \\ 10& 15\end{array}\right]$$

$$\left[\begin{array}{cc}50& 45\\ \\ 60& 45\\ \\ 20& 30\end{array}\right]$$

$$A2$$

$$B2$$

$$C2$$

Shop#2 is cheaper. Drag peasant#3 and go there, peasants#1 and #2.

How did this multiplication happen? Same story as with one shop/prices, repeated to handle the second shop. Matrix B2 now has two columns for prices, for Shop#1 and Shop#2. Multiplication takes each column in B2 and does the same weighted-column-addition with A2. Hence two columns of answers in C2.

ImportantMatrix Multiplication in 4 ways

As shown in Gilbert Strang’s book, there are 4 ways of thinking about matrix multiplication:

• Our method of weighting columns in A using columns B and adding
• Weighting rows in B using rows in A and adding
• Multiplying individual values from rows in A and columns in B and adding up, for each location in C. ( This is the common textbook method)
• Multiplying rows in A and columns in B. ( A vectorized version of the previous method).

For now, this one method should suffice.

References

1. Gilbert Strang. Linear Algebra and its Applications. Thomson/Brooks-Cole.
2. Glenn Henshaw.(Sep 28, 2019) Three Ways to Understand Matrix Multiplication. https://ghenshaw-work.medium.com/3-ways-to-understand-matrix-multiplication-fe8a007d7b26