# mpower, ^

## Syntax

``C = A^B``
``C = mpower(A,B)``

## Description

example

````C = A^B` computes `A` to the `B` power and returns the result in `C`.```
````C = mpower(A,B)` is an alternate way to execute `A^B`, but is rarely used. It enables operator overloading for classes.```

## Examples

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Create a 2-by-2 matrix and square it.

```A = [1 2; 3 4]; C = A^2```
```C = 2×2 7 10 15 22 ```

The syntax `A^2` is equivalent to `A*A`.

Create a 2-by-2 matrix and use it as the exponent for a scalar.

```B = [0 1; 1 0]; C = 2^B```
```C = 2×2 1.2500 0.7500 0.7500 1.2500 ```

Compute `C` by first finding the eigenvalues `D` and eigenvectors `V` of the matrix `B`.

`[V,D] = eig(B)`
```V = 2×2 -0.7071 0.7071 0.7071 0.7071 ```
```D = 2×2 -1 0 0 1 ```

Next, use the formula `2^B = V*2^D/V` to compute the power.

`C = V*2^D/V`
```C = 2×2 1.2500 0.7500 0.7500 1.2500 ```

## Input Arguments

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Operands, specified as scalars or matrices. Inputs `A` and `B` must be one of the following combinations:

• Base `A` and exponent `B` are both scalars, in which case `A^B` is equivalent to `A.^B`.

• Base `A` is a square matrix and exponent `B` is a scalar. If `B` is a positive integer, the power is computed by repeated squaring. For other values of `B` the calculation uses an eigenvalue decomposition (for most matrices) or a Schur decomposition (for defective matrices).

• Base `A` is a scalar and exponent `B` is a square matrix. The calculation uses an eigenvalue decomposition.

Operands with an integer data type cannot be complex.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `logical` | `char`
Complex Number Support: Yes

## Tips

• MATLAB® computes `X^(-1)` and `inv(X)` in the same manner, and both are subject to the same limitations. For more information, see `inv`.

## References

[1] Higham, Nicholas J., and Lijing Lin. “An Improved Schur--Padé Algorithm for Fractional Powers of a Matrix and Their Fréchet Derivatives.” SIAM Journal on Matrix Analysis and Applications 34, no. 3 (January 2013): 1341–1360. https://doi.org/10.1137/130906118.

## Version History

Introduced before R2006a

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