# Tagged Questions

**7**

votes

**1**answer

163 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

**1**answer

210 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

**1**answer

116 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

**0**answers

119 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

**0**answers

164 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

**1**answer

302 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

**2**answers

841 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

**1**answer

255 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 ...

**2**

votes

**2**answers

1k 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

**0**answers

294 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

**1**answer

192 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 ...