POWER 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 = 
  <double>  - size: [1 1]
 1.5  
--> B = [1,.2;.2,1]
B = 
  <double>  - size: [2 2]
 
Columns 1 to 2
 1.0  0.2  
 0.2  1.0  

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

--> C = B^A
C = 
  <double>  - size: [2 2]
 
Columns 1 to 2
 1.0150379454061658  0.2994961926062329  
 0.2994961926062330  1.0150379454061658  

Next, we raise A to the matrix power B:

--> C = A^B
C = 
  <double>  - size: [2 2]
 
Columns 1 to 2
 1.50493476200956966  0.12177289478697813  
 0.12177289478697809  1.50493476200956966