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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')