Creating a matrix with desirable eigenvalues

[Deimantas]

Hello,

I want to generate a (large) matrix with eigenvalues that are all in a small interval. The relationship between the maximum eigenvalue and minimum eigenvalue should be as small as possible, that's the goal. And the eigenvalues must all be positive.

Is there any simple way to do this? I'm using Mathcad, and I've built a function to randomly generate a symmetric matrix. And when I want all the eigenvalues positive, I just use the matrix multiplied by it's transpose. But the eigenvalues are scattered in a large interval, and the relationship between the max eigenvalue and min eigenvalue is in the hundreds and thousands. Any tricks to make the eigenvalues get closer to each other?

 

[Office_Shredder]

Are you trying to generate random matrices? Because you can just pick a diagonal matrix if all you want is an arbitrary single such matrix.

Otherwise pick a diagonal matrix randomly, and conjugate it by a random orthogonal transformation. The eigenvalues are determined by the diagonal entries and the eigenvectors by the orthogonal matrix so by picking them with the appropriate distribution you can get your eigenvectors and eigenvalues to have whatever distribution you want without too much fuss.

 

[Deimantas]

Are you trying to generate random matrices? Because you can just pick a diagonal matrix if all you want is an arbitrary single such matrix.

Otherwise pick a diagonal matrix randomly, and conjugate it by a random orthogonal transformation. The eigenvalues are determined by the diagonal entries and the eigenvectors by the orthogonal matrix so by picking them with the appropriate distribution you can get your eigenvectors and eigenvalues to have whatever distribution you want without too much fuss.

Thanks for replying. So that's the easiest way, right? Could you give a simple example of conjugating a small diagonal matrix by an orthogonal transformation?

 

[hilbert2]

^ Let's say you want a 2x2 matrix with eigenvalues 2 and 3. First you form the diagonal matrix

$\begin{pmatrix}2&0\\0&3\end{pmatrix}$

then you multiply this matrix from the left side with the rotation matrix (which is orthogonal)

$\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}$

and finally you multiply the resulting matrix from the right side with the inverse rotation matrix

$\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$

If the parameter $\theta$ is chosen randomly from interval $[0,2\pi]$, the eigenvalues are probably not obviously visible from the resulting matrix, but they will still be 2 and 3.

 

[Deimantas]

Thank you both for the help, much appreciated

 

[AlephZero]

Look at the test programs that come with the LAPACK library, and their documentation. There are matrix generation routines that do this type of thing, and for "large" matrices they will probably have better numerical behavior than code you invent for yourself. IIRC they use other numerical methods besides the idea in post #4. http://www.netlib.org/lapack/

 

[Deimantas]

I'm afraid acquinting myself with LAPACK would take too much time.. Oh and it turns out matrix rotation isn't exactly the thing I was looking for, because what I need is a 100x100 matrix, not 2x2 or 3x3..

 

[hilbert2]

Oh and it turns out matrix rotation isn't exactly the thing I was looking for, because what I need is a 100x100 matrix, not 2x2 or 3x3..

You don't have to use a rotation matrix as the orthogonal transformation. You can form other orthogonal transformations by making NxN matrices that have a set of N orthogonal vectors as their columns. If you want to make a set of N orthogonal vectors in ℝ^N, just create N random vectors and use the Gram-Schmidt orthogonalization method on them.

Once you have created the orthogonal matrix $U$ that has the random orthogonal vectors as its columns, you just make the diagonal matrix $D$ that has the desired eigenvalues, and then compute $U^{-1}DU$, which is a random matrix that has the eigenvalues you want. Remember that for an orthogonal matrix $U$, $U^{-1}$ is just the transpose of $U$.

For more info, see

http://en.wikipedia.org/wiki/Orthogonal_matrix

http://en.wikipedia.org/wiki/Gram-schmidt