^^ (Matrix Power)

Purpose:

Raises a matrix to a scalar power or a scalar to a matrix power.

Syntax:

A ^^ p

A	-	A non-singular, square matrix or a scalar. The base.
p	-	A square matrix or scalar if A is a matrix. The exponent.

Returns:

A matrix.

Example:

a = {{1, 2, 3},

{4, 5, 6},

{7, 8, 0}}

b = a^^3

b == {{279, 360, 306},

{684, 873, 684},

{738, 900, 441}}

c = a^^1.5

c == {{ 7.245 - 2.116i, 9.115 - 1.718i, 6.304 + 2.891i},

{17.096 - 2.596i, 21.508 - 2.608i, 14.875 + 4.099i},

{15.465 + 6.185i, 19.456 + 5.746i, 13.457 - 9.250i}}

d = 1.5^^a

d == {{24.009, 29.166, 20.401},

{54.739, 69.769, 47.985},

{49.996, 62.783, 43.550}}

Remarks:

A^^0 == eye(size(A)) the identity matrix.

A^^-1 == inv(A) the matrix inverse.

A^^1 == A

Given matrix A and scalar p:

For p a positive integer, A^^p is equivalent to:

A *^ A ^* A *^ A ... *^ A (p times.)

For p a negative integer, A^^p is equivalent to the positive case except the inverse of A is multiplied.

For p a real or complex number, the function computes the matrix power using eigenvectors and eigenvalues as follows:

(v, d) = eig(A)

A^^p == (v *^ (d ^ p)) /^ v

For p^^A, the matrix power is calculated as:

(v, d) = eig(A)

(v *^ diag(p ^ diag(d))) /^ v

If neither p nor A are matrices, ^^ defaults to the standard ^ operator. For example:

4 ^^ 3 == 64

If both p and A are matrices, an error occurs.

See the ^ operator to raise each element of an array to a power.