Mathematics

Matrix Powers and Exponentials

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

• Positive integer
• Inverse and fractional
• Element-by-element
• Exponentials

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.

• 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 = B^(-3)
  
  Y =
      0.0053   -0.0068    0.0018
     -0.0034    0.0001    0.0036
     -0.0016    0.0070   -0.0051
  

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

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
  

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