# 08. Other operations

## 08. Other operations

In this video, we'll define some further operations you can do to produce new matrices. The first is matrix addition If we have two m-by-n matrices A and B with entries A_{i j} and B_{i j}, we can form a new matrix A + B with (A + B)_{i j} = A_{i j} + B_{i j}. In other words, you take the i jth entries of both matrices and add them.

the 2-by-2 matrix 1, 0; 1, 1 plus the 2-by-2 matrix 1, 1; 0, minus 1 equals the 2-by-2 matrix 2, 1; 1, 0.

This is most useful when A and B are both column vectors, i.e. m-by-1 matrices. Let's see what it means in for vectors in R 2. The formula is x, y plus a, b equals x + a, y + b

Geometrically, we add two vectors v = x, y and w = a, b by translating w to the tip of v and drawing the arrow from the tail of v to the tip of w. One can see from the picture that the x- (respectively y-) coordinate of this arrow is the sum of the x- (respectively y-) coordinates of v and w.

### Rescaling

Given a number lambda and a matrix A, you can form the matrix \lambda A whose entries are lambda times the entries of A.

2 times the 2-by-2 matrix 1, 2; 3, 4 equals the 2-by-2 matrix 2, 4; 6, 8.

### Matrix exponentiation

The exponential of a number x is defined by the Taylor series of exp: exp of x equals 1 plus x plus x squared over 2 factorial plus x cubed over 3 factorial plus dot dot dot, which equals the sum over n from 0 to infinity of x to the n over n factorial. We can use the same definition to define the exponential of a matrix: exp of M equals the sum over n from 0 to infinity of M to the n over n factorial. Here, A to the zero is understood to mean the identity matrix I (the analogue for matrices of the number 1).

Consider M, the 2-by-2 matrix 0, 1; 0, 0. Since M squared equals zero, all the higher powers of M vanish (the name for this is nilpotence: some power of M is zero), so the matrix exponential becomes exp of M equals I plus M, which equals 1, 1; 0, 1. So we get the matrix for a shear as the exponential of a nilpotent matrix.

In fact, exp of 0, t; 0, 0 equals 1, t; 0, 1, so we get a whole family of matrices which shear further and further to the right as t varies.

Take M to be the 2-by-2 matrix 0, minus t; t, 0. We have M equals t times 0, minus 1; 1, 0, M squared equals minus t^2, 0; 0, minus t squared, which equals minus t squared times the identity, M cubed equals minus t cubed times 0, minus 1; 1, 0, M to the four equals minus t to the four times the identity etc.

and in the end we get exp of M equals the identity plus t times 0, minus 1; 1, 0, minus t squared over 2 times the identity, minus t cubed over 3 factorial times 0, minus 1; 1, 0, plus t to the 4 over 4 factorial times the identity, plus t to the 5 over 5 factorial times 0, minus 1; 1, 0, etc. The coefficient of I is the Taylor series for cos t; the coefficent of 0, minus 1; 1, 0 is the Taylor series for sine t, so overall we get exp of M equals cos t, minus sine t; sine t, cos t. So we get a general rotation matrix in 2-d by exponentiating this very simple matrix.