7
votes
1answer
132 views

How much can I perturb a symmetric stochastic matrix and keep positive solutions?

Suppose $A$ is a symmetric stochastic $n \times n$ matrix with least eigenvalue $\lambda_n>0$. Consider real symmetric perturbations $\widehat{A}$. How large can I take $\epsilon$ such that ...
5
votes
1answer
202 views

Separating the spectrum of a Hermitian matrix

Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated. Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each ...
4
votes
1answer
105 views

Distribution of the spectrum of a perturbed matrix

Let $A$ be an $n\times n$ Hermitian matrix, with well-separated eigenvalues $\lambda_1 > \lambda_2 ... > \lambda_n$, with $|\lambda_i-\lambda_j|>\epsilon$, for all $i \neq j$. Let $G$ be a ...
2
votes
0answers
101 views

Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...
1
vote
0answers
156 views

componentwise eigenvector perturbation

Does the sin-theta theorem imply a componentwise nonasymptotic bound for eigenvectors? Assume, for the purpose of this question, that the eigenvalues concerned are simple. If this is trivial, I ...
2
votes
1answer
286 views

Null Space Perturbations

Hi, I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases. The distilled version of the ...
4
votes
2answers
755 views

rank-one perturbation of a matrix corresponding to a specific spectrum

Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,...,\lambda_n$ Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
3
votes
1answer
252 views

enlarge the separation between two matrices

The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as $$ \operatorname{sep}(A,B)=\min_{X\neq ...
1
vote
2answers
923 views

submultiplicity of matrix norm. Is $||AB||_F \leq ||A||_2||B||_F$?

Let $||\cdot||_F$ and $||\cdot||_2$ be the Frobenius norm and the spectral norm. I'm reading Ji-Guang Sun's paper 'Perturbation Bounds for the Cholesky and QR Factorizations' from BIT 31 in 1991. ...
3
votes
0answers
284 views

A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede

In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values) Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq ...
7
votes
1answer
185 views

Adding a multiple of the Identity to a LU factorized matrix

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the ...