Section: Mathematical Operators
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
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
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