All Questions
11 questions with no upvoted or accepted answers
13
votes
0
answers
809
views
Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it
In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
- 673
4
votes
0
answers
989
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
3
votes
0
answers
130
views
The probability that the dominant eigenvalue of a random real matrix is real
Let $X_n$ be an $n\times n$ real matrix where the entries in $X_n$ are independent, normally distributed, have mean $0$, and variance $1$. Suppose that $\lambda_1,\dots,\lambda_n$ are the eigenvalues ...
2
votes
0
answers
172
views
Minimum of $\mathrm{rank}\left( \boldsymbol{W} \boldsymbol{H} \right)$, with $\boldsymbol{W}$ block diagonal
Let us assume that we have a full-rank $(n\cdot l)\times k$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), and an $m\times (n\cdot l)$ ...
- 61
2
votes
0
answers
102
views
Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries
I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
- 121
1
vote
0
answers
19
views
Empirical approaches to validate observational bounds on minimum gap between least eigenvalues of $n \times n$ correlation matrix and its submatrices
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an $(n-1) \times (n-1)$ submatrix of $\Sigma$ obtained by eliminating the $i$-th row ...
- 141
1
vote
0
answers
112
views
Tools to bound the singular values of a finite sum of random matrices from below?
Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
- 111
1
vote
0
answers
46
views
the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?
Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance matrix,...
- 275
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
0
votes
0
answers
47
views
"Probability" for a partitioned matrix to be singular
Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
- 2,712
0
votes
0
answers
704
views
expected matrix inverse of circulant plus diagonal matrix with chi-square variables
Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$.
Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...
- 96