power (^,.^) power operation

## Syntax

    t=A^b
                t=A**b
                t=A.^b

## Arguments

A,t

  scalar, polynomial or rational matrix.

b

  a scalar, a vector or a scalar matrix.

## Description

- If `A` is a square matrix and `b` is a scalar then `A^b` is the matrix `A` to the power `b`.

- If `b` is a scalar and `A` a matrix then `A.^b` is the matrix formed by the element of `A` to the power `b` (element-wise power). If `A` is a vector and `b` is a scalar then `A^b` and `A.^b` performs the same operation (i.e. element-wise power).

- If `A` is a scalar and `b` is a square matrix `A^b` is the matrix `expm(log(A) * b)`.

  If `A` is a scalar and `b` is a vector `A^b` and `A.^b` are the vector formed by `a^(b(i,j))`.

  If `A` is a scalar and `b` is a matrix `A.^b` is the matrix formed by `a^(b(i,j))`.

- If `A` and `b` are vectors (matrices) of the same size `A.^b` is the `A(i)^b(i)` vector (`A(i,j)^b(i,j)` matrix).

## Additional Remarks

Notes:

1. For square matrices `A^p` is computed through successive matrices multiplications if `p` is a positive integer, and by diagonalization if not (see "note 2 and 3" below for details).

2. If `A` is a square and Hermitian matrix and `p` is a non-integer scalar, `A^p` is computed as:

`A^p = u*diag(diag(s).^p)*u'` (For real matrix `A`, only the real part of the answer is taken into account).

`u` and `s` are determined by `[u,s] = schur(A)` .

3. If `A` is not a Hermitian matrix and `p` is a non-integer scalar, `A^p` is computed as:

`A^p = v*diag(diag(d).^p)*inv(v)` (For real matrix `A`, only the real part of the answer is taken into account).

`d` and `v` are determined by `[d,v] = bdiag(A+0*%i)` .

4. If `A` and `p` are real or complex numbers, `A^p` is the principal value determined by:

`A^p = exp(p*log(A))` (or `A^p = exp(p*(log(abs(A))+ %i*atan(imag(A)/real(A))))` ).

5. If `A` is a square matrix and `p` is a real or complex number, `A.^p` is the principal value computed as:

`A.^p = exp(p*log(A))` (same as case 4 above).

6. `**` and `^` operators are synonyms.

Exponentiation is right-associative in Scilab contrarily to Matlab® and Octave. For example 2^3^4 is equal to 2^(3^4) in Scilab but is equal to (2^3)^4 in Matlab® and Octave.

## Examples

    A=[1 2;3 4];
    A^2.5,
    A.^2.5
    (1:10)^2
    (1:10).^2

    A^%i
    A.^%i
    exp(%i*log(A))

    s=poly(0,'s')
    s^(1:10)

## See also

- exp
- hat