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