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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')
```