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.

Inverse and Fractional Powers

If A is square and nonsingular, then A^(-p) effectively multiplies inv(A) by itself p-1 times.

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,

Exponentials

The function

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

computes the matrix exponential. An example is provided by the 3-by-3 coefficient matrix

and the initial condition, x(0)

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

A three-dimensional phase plane plot obtained with

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