Matrix Powers and Exponentials :: Matrices and Linear Algebra (Mathematics)

Mathematics

Matrix Powers and Exponentials

This section tells you how to obtain the following matrix powers and exponentials in MATLAB:

Positive Integer Powers

If A is a square matrix and p is a positive integer, then 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, then 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

derivative of x with respect to t = A times x

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,

x(t) = e to the t times A power times x(0)

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

and the initial condition, x(0)

```
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 0 t 1 with

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

A three-dimensional phase plane plot obtained with

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

shows the solution spiraling in towards the origin. This behavior is related to the eigenvalues of the coefficient matrix, which are discussed in the next section.

QR Factorization Eigenvalues