How to generate random matrices for given eigenvalues in matlab?

Question

Is there an easy way to generate random matrices, all with eigenvalues that I'll choose?

Answer 1

Given a diagonal matrix containing all the eigenvalues you want, you can use a similarity transformation of a random (but invertible) matrix to do what you want:

>> lambda = [-1 -2 -3]   % These are the eigenvalues you want
lambda =
  -1    -2    -3

>> A = diag(lambda)
A =
-1     0     0
 0    -2     0
 0     0    -3

>> eig(A)
 ans =
 -3
 -2
 -1

>> S = rand(size(A));   % Random matrix of compatible size with A.
>> B = inv(S)*A*S       % Random matrix with desired eigenvalues, using similarity transformation
B =
-2.7203   -0.5378   -0.9465
 2.0427   -0.1766    1.2659
-2.0817   -1.8974   -3.1031

>> eig(B)
ans =
-3.0000
-1.0000
-2.0000