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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

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.

The `.^`

operator produces element-by-element
powers. For example:

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

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')
```