FreeMat: MATRIXPOWER Matrix Power Operator

FreeMat

- Main Page
- Related Pages

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

# 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