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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