MATRIXPOWER Matrix Power Operator

Section: Mathematical Operators

Usage

The power operator for scalars and square matrices. This operator is really a combination of two operators, both of which have the same general syntax: 
  y = a ^ b

The exact action taken by this operator, and the size and type of the output, depends on which of the two configurations of a and b is present:

  1. a is a scalar, b is a square matrix
  2. a is a square matrix, b is a scalar

Function Internals

In the first case that a is a scalar, and b is a square matrix, the matrix power is defined in terms of the eigenvalue decomposition of b. Let b have the following eigen-decomposition (problems arise with non-symmetric matrices b, so let us assume that b is symmetric):

Then a raised to the power b is defined as

Similarly, if a is a square matrix, then a has the following eigen-decomposition (again, suppose a is symmetric):

Then a raised to the power b is defined as

Examples

We first define a simple 2 x 2 symmetric matrix 
--> A = 1.5

A = 
    1.5000 

--> B = [1,.2;.2,1]

B = 
    1.0000    0.2000 
    0.2000    1.0000

First, we raise B to the (scalar power) A: 

--> C = B^A

C = 
    1.0150    0.2995 
    0.2995    1.0150

Next, we raise A to the matrix power B: 

--> C = A^B

C = 
    1.5049    0.1218 
    0.1218    1.5049