Symbolic spectra radius of matrix?

General Usage
1

I’m looking for the spectral radius of matrix M given by:

M =\left(\begin{array}{ccc}0 & k & r\\1 & 0 & 0\\0 & 1-n & 0\end{array}\right)

where all of k,r,n are non-negative reals.

WolframAlpha can compute the eigenvalues: there are one real and two complex conjugate eigenvalues. But the expressions are really ugly.

Does any of the Julia symbolic packages support spectral radius computations?

2

SymPy can handle this 3x3 case:

using SymPy, LinearAlgebra
@vars k r n positive=true
M = [0 k r; 1 0 0; 0 1-n 0]
eigvals(M)
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3

Can I also specify that 0 < n < 1?

4

You’re solving a cubic polynomial, so the analytical expressions are always going to be ugly, no?

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5

I suspect the expressions will be ugly (as you say). The problem is that I try to recreate an expression in a publication, and the expression in the publication is really, really simple.

Of course, there is always the possibility that the expression in the publication is incorrect…

6

You shoud be able to verify the solution in the paper directly (or show that it’s wrong), by substituting it into \det(A-\lambda I)=0.

4 Likes
7

Good point.

On second thought… [corrected after tip from dpsanders]

\rho(A) =\max_i |\lambda_i (A)|

so I guess there is no guarantee that the spectral radius is an eigenvalue?

8

That should be max, not argmax. But yes, it’s the modulus of an eigenvalue.

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9

Yes, of course… the arg max would give the i…