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FreeMat
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Section: Mathematical Operators
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 =
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