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
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
We first define a simple 2 x 2 symmetric matrix
--> A = 1.5
A =
<double> - size: [1 1]
1.500000000000000
--> B = [1,.2;.2,1]
B =
<double> - size: [2 2]
Columns 1 to 2
1.000000000000000 0.200000000000000
0.200000000000000 1.000000000000000
First, we raise B to the (scalar power) A:
--> C = B^A
C =
<double> - size: [2 2]
Columns 1 to 2
1.015037945406166 0.299496192606233
0.299496192606233 1.015037945406166
Next, we raise A to the matrix power B:
--> C = A^B
C =
<double> - size: [2 2]
Columns 1 to 2
1.504934762009570 0.121772894786978
0.121772894786978 1.504934762009570