All Questions
36 questions
0
votes
0
answers
66
views
Random elliptical potential lemma
Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
- 61
36
votes
4
answers
2k
views
Determinant of the random matrix $X^2+Y^2$
$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$.
i) When $n=2,3,4$, one ...
- 3,741
8
votes
1
answer
323
views
On a matrix inequality
$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,
$$\...
- 128k
4
votes
0
answers
196
views
What is the minimum nonzero rank in a random subspace of matrices?
Fix positive integers $m$, $n$, and $k\leq mn$, and draw a $k$-dimensional subspace $S\leq\mathbb{R}^{m\times n}$ uniformly from the Grassmannian.
What is known about the random variable
$R(m,n,k):=\...
- 7,633
2
votes
0
answers
181
views
Is every nearly rank-1 doubly stochastic matrix a product of pairwise averaging matrices?
A doubly stochastic matrix is a square matrix with non-negative real entries where the sum of each row is $1$ and the sum of each column is $1$. A pairwise averaging matrix is a matrix of the form $tA+...
2
votes
1
answer
236
views
How can I prove a randomly generated matrix has distinct non-zero eigenvalues?
Consider the following $M×M$ matrix
$$
\mathbf A=\sum_{k=1}^K =a_k \mathbf h_k \mathbf h_k^H,(M≥K)
$$
where $a_k$'s are real values and $h_k$'s are $M×1$ randomly generated vectors, e.g., complex ...
- 31
1
vote
2
answers
142
views
If $x \ge 0$ and $\mathbf{1}^Tx \le \|x\|^2$ then $\mathbf{1}^T(I - xx^T / \|x\|^2) \mathbf{1} \ge \| [\mathbf{1} - x]_+ \|^2$
Notation. Denote $\mathbf{1}=(1,1,\ldots,1)$ as the vector-of-ones in $\mathbb{R}^n$. Write the "positive part" as $[\alpha]_+ = \max\{\alpha,0\}$ for $\alpha\in\mathbb{R}$ and $[(x_1,x_2,\...
- 570
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
0
votes
0
answers
47
views
"Probability" for a partitioned matrix to be singular
Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
- 2,712
2
votes
0
answers
326
views
Explicit formula for this distance between positive semi-definite matrices?
Let $A$ and $B$ in $\mathbb{R}^{d\times d}$ be positive semi-definite (psd) matrices and let $d\tau$ be the uniform probability distribution on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$. I ...
1
vote
1
answer
218
views
Is there a bound on the norm of the product of second moment matrix with random vector?
Let $X_1,\dots,X_n$ be vectors in $\mathbb{R^d}$. Assume all of the vectors are inside the unite $\ell_2$ ball, but outside the ball of radius $r$ for some $r \in (0,1)$, i.e. $r \leq \|X_i\| \leq 1$ ....
- 135
2
votes
1
answer
280
views
Properties of eigenvalues and eigenvectors of a particular random matrix
Let $\mathbf{A}$ be a given $n \times m$ matrix with positive entries, and $\mathbf{B}_{n\times m}$ be a random i.i.d complex Gaussian matrix with unit variance. Assume that $\mathbf{C}$ is the ...
- 287
6
votes
2
answers
738
views
Probability of a large random integer Matrix to have zero determinant
Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$
I would like to ...
- 355
4
votes
2
answers
932
views
Steady state Kalman filter
My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from.
Kalman filter allows us to estimate state at time $t$ as ...
- 67
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
3
votes
0
answers
419
views
(Expected) Size of smallest singular value of a Vandermonde matrix associated to roots of polynomial
Let $n,H$ two fixed positive integers.
Let $P\in\mathbb{Z}[X]$ a monic integral polynomial of height $H$ and degree $n$ taken uniformly at random (i.e. each of the $n$ free coefficients of $P$ is ...
- 313
3
votes
0
answers
182
views
Spectral radius of infinite substochastic upper triangular matrix
Let $M$ be a Markov chain on $\{0, 1, 2, \dots\} \cup \{\delta\}$, where $\Pr(i \to j) > 0$ for $i, j \in \mathbb{N}$ only if $j > i$, and $\Pr(\delta \to \delta) = 1$. This represents a birth-...
- 131
1
vote
1
answer
394
views
On rank of random $0/1$ matrices
It is known that a $0/1$ matrix picked from uniform distribution from $\{0,1\}^{n\times n}$ is non-singular with probability $1-o(1)$.
Fix an integer $t$.
Consider a random matrix formed the ...
user94040
1
vote
0
answers
167
views
Expected amount of linearly dependent random vectors? [closed]
Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$?
EDIT: As said in the comments, I'm looking for the ...
- 123
3
votes
1
answer
371
views
Eigenvectors of a perturbed reducible stochastic matrix
Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix
$$\tilde{Q}\,=\,(1-\alpha)...
- 1,421
-2
votes
3
answers
447
views
Determinant of matrix from set {-1, 1} [closed]
Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have ...
- 45
27
votes
3
answers
13k
views
What is known about the distribution of eigenvectors of random matrices?
Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...
- 433
7
votes
1
answer
355
views
Injectivity of matrix "fingerprint"
Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries.
For any matrix $A$, define
$$ D_A(i,j)=\...
- 3,979
4
votes
1
answer
189
views
Weak ergodicity of nonhomogenous products of 0-1 matrices
Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...
- 4,679
12
votes
0
answers
825
views
Eigenvalues of permutations of a real matrix: how complex can they be?
This is sort of complementary to this thread. I’ll repeat the definitions here:
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
- 13.4k
22
votes
4
answers
5k
views
Eigenvalues of permutations of a real matrix: can they all be real?
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
- 13.4k
3
votes
0
answers
549
views
Canonical forms for block-positive-definite matrices
Suppose we are given a block $2\times 2$ matrix that is positive-definite, and let's suppose for simplicity that the blocks along the main diagonal are the identity. So
$$
\begin{bmatrix} I & X \\\...
- 427
42
votes
3
answers
5k
views
The probability for a symmetric matrix to be positive definite
Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
- 52.3k
2
votes
1
answer
719
views
Lower bound on Bhattacharya distance between independent Gaussian distributions ?
I am interested in a lower bound on the Bhattacharya distance between two independent multivariate Gaussian distributions. To be precise, consider zero-mean independent Gaussian distributions $p_1\sim\...
- 163
14
votes
1
answer
1k
views
A Question on Random Matrices
Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by
$$
V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q})
$$
where $x_{1},x_{2},\ldots,x_{n}$ are iid ...
- 3,626
15
votes
3
answers
4k
views
Non-diagonalizable doubly stochastic matrices
Are there constructive examples of doubly stochastic matrices (whose rows and columns all sum up to $1$ and contain only non-negative entries) that are not diagonalizable?
- 731
10
votes
1
answer
1k
views
Bounds on $\|P^{k+1} - P^k\|$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k\gg n$
The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...
- 303
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181
29
votes
3
answers
3k
views
Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 2,200
5
votes
2
answers
2k
views
Matrices whose exponential is stochastic
The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
- 1,532
18
votes
3
answers
8k
views
Number of invertible {0,1} real matrices?
This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices ...
- 32.1k