Leading eigenpair of degenerate non-symmetric matrix

Asked Jul 03 '23 at 01:16

Active Jul 03 '23 at 07:38

Viewed 35 times

Consider a non-symmetric matrix ${\bf A}_0$ with one eigenvalue $\lambda$ and all other eigenvalues are zero. The corresponding left $\bf u$ and right $\bf v$ eigenvectors to the eigenvalue $\lambda$ are normalised using the following relation: ${\bf u} \cdot {\bf v} = 1$.

Assume I perturb ${\bf A}_0$ with a small perturbation such that

$$ {\bf A} = {\bf A}_0 + \epsilon {\bf A}_1 $$

I know how to calculate the first order correction in the leading eigenvalue $\lambda$. However, what is the first order correction of the leading eigenvectors?

edited Jul 03 '23 at 07:38

Rodrigo de Azevedo

asked Jul 03 '23 at 01:16

Matt

If ${\bf A}_0$ is rank-1 — is it? — why not replace it with something like $\bf a b^\top$? – Rodrigo de Azevedo Jul 03 '23 at 07:30
It is rank 1 indeed, I can write $A_0$ as $\mathbf{ab}^\mathsf{T}$. Forgive my lack of understanding: how would this help me to find the first order perturbation in the eigenvectors? – Matt Jul 03 '23 at 07:47
1

Then you have a rank-1 update of $\epsilon {\bf A}_1$, right? Take a look at this – Rodrigo de Azevedo Jul 03 '23 at 07:55
Take a look at these – Rodrigo de Azevedo Jul 03 '23 at 08:01
Is ${\bf A}_0 = \lambda {\bf u} {\bf v}^\top$? – Rodrigo de Azevedo Jul 03 '23 at 08:03
Yes it is, does this make the problem simple to solve? – Matt Jul 03 '23 at 08:24
Use the matrix determinant lemma to compute the characteristic polynomial. Take a look at this – Rodrigo de Azevedo Jul 03 '23 at 08:29
I had a look at the links. My current obstacles are that 1) I do not know the eigenvalues/vectors of $\mathbf{A}_1$; 2) Neither $\mathbf{A}_0$ nor $\mathbf{A}_1$ are symmetric, so I have difficulties applying a rank-1 update of $\varepsilon\mathbf{A}_1$. Finally, the matrix determinant lemma looks intersting, but I do not know how I can obtain an expression for the leading eigenvector from it? – Matt Jul 04 '23 at 01:35

Leading eigenpair of degenerate non-symmetric matrix

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