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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

$$ dx/dt = Ax, $$

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

$$ x(t) = e^{tA} x(0) $$ .

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]
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 $0 \le t \le 1.$

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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