# Documentation

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## Powers and Exponentials

### Positive Integer Powers

If `A` is a square matrix and `p` is a positive integer, `A^p` effectively multiplies `A` by itself `p-1` times. For example:

```A = [1 1 1;1 2 3;1 3 6] A = 1 1 1 1 2 3 1 3 6 X = A^2 X = 3 6 10 6 14 25 10 25 46```

### Inverse and Fractional Powers

If `A` is square and nonsingular, `A^(-p)` effectively multiplies `inv(A)` by itself `p-1` times:

```Y = A^(-3) Y = 145.0000 -207.0000 81.0000 -207.0000 298.0000 -117.0000 81.0000 -117.0000 46.0000```

Fractional powers, like `A^(2/3)`, are also permitted; the results depend upon the distribution of the eigenvalues of the matrix.

### Element-by-Element Powers

The `.^` operator produces element-by-element powers. For example:

```X = A.^2 A = 1 1 1 1 4 9 1 9 36```

### Exponentials

The function

```sqrtm(A) ```

computes `A^(1/2)` by a more accurate algorithm. The `m` in `sqrtm` distinguishes this function from `sqrt(A)`, which, like `A.^(1/2)`, does its job element-by-element.

A system of linear, constant coefficient, ordinary differential equations can be written

where x = x(t) is a vector of functions of `t` and `A` is a matrix independent of `t`. The solution can be expressed in terms of the matrix exponential

.

The function

```expm(A) ```

computes the matrix exponential. An example is provided by the 3-by-3 coefficient matrix,

```A = [0 -6 -1; 6 2 -16; -5 20 -10] ```
```A = 0 -6 -1 6 2 -16 -5 20 -10 ```

and the initial condition, x(0).

```x0 = [1 1 1]' ```
```x0 = 1 1 1 ```

The matrix exponential is used to compute the solution, x(t), to the differential equation at 101 points on the interval .

```X = []; for t = 0:.01:1 X = [X expm(t*A)*x0]; end ```

A three-dimensional phase plane plot shows the solution spiraling in towards the origin. This behavior is related to the eigenvalues of the coefficient matrix.

```plot3(X(1,:),X(2,:),X(3,:),'-o') ```

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