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:

a
is a scalar,b
is a square matrix 
a
is a square matrix,b
is a scalar
Function Internals
In the first case thata
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 eigendecomposition (problems arise with nonsymmetric 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 eigendecomposition (again, suppose a
is symmetric):
Then a
raised to the power b
is defined as
Examples
We first define a simple2 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