# The matrix exponential

## The matrix exponential function

Definition (The matrix exponential):

If M is a square matrix then the exponential of M is defined to be exp of M equals the identity, plus M, plus a half M squared, plus one over 3 factorial M cubed plus dot dot dot, i.e. the sum from n equals 0 to infinity of one over n factorial M to the n. Here M to the zero means the identity.

Example:

Let M be the 2-by-2 matrix 0, minus theta, theta, 0. We'll compute \exp(M). First let's compute the powers of M:

• M squared equals minus theta squared, 0, 0 minus theta squared, which is just minus theta squared times the identity matrix,

• M cubed equals 0, theta cubed, minus theta cubed, 0,

• M to the 4 equals (M squared) squared, which is theta to the 4 times the identity.

• M to the 5 equals theta to the 4 times M, etc

so we end up with: exp M equals the 2-by-2 matrix whose entries are: 1 minus a half theta squared plus one over 4 factorial theta to the 4 dot dot dot, minus theta plus one over three factorial theta cubed minus dot dot dot, theta minus one over three factorial theta cubed plus dot dot dot, 1 minus a half theta squared plus one over 4 factorial theta to the four dot dot dot, which is just the 2-by-2 matrix cos theta, minus sine theta, sine theta, cos theta where we have just observed that the power series in each entries are the Taylor series of cos and sin.

Remark:

This is the 2-by-2 matrix that gives you a rotation by an angle \theta. You should imagine that, as in this example, the exponential map "eats" a very simple matrix (like an n-by-n antisymmetric matrix) and outputs a much more complicated and useful matrix (like an n-by-n rotation matrix).

## Pre-class exercise

Exercise:

Compute exp of the 2-by-2 matrix 0, x, 0, 0 and exp of the 3-by-3 matrix 0, a, c, 0, 0, b, 0, 0, 0.