All Questions
Tagged with matrices pr.probability
44 questions with no upvoted or accepted answers
21
votes
0
answers
2k
views
The Fourier Transform of taking Eigenvalues
The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
- 24.7k
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
9
votes
0
answers
802
views
Positive definiteness of matrix
This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows:
We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
- 192
8
votes
0
answers
254
views
Quantum coupon collection: positivity of an alternating sum of matrices
It is well-known that in the classic coupon collecting problem (CCP), the expected waiting time is
\begin{equation*}
T_n(x_1,\ldots,x_n) = \sum_{k=1}^n (-1)^{k+1}\sum_{1\le i_1 < \cdots < i_k \...
- 28.6k
5
votes
0
answers
352
views
0-1 matrix combinatorial problem
Let $M \in \{0,1\}^{n \times n}$ and let $r_i$ be its $i$-th row. Given constant $p \in (0,1/2]$, let the number of $1$s in each row be at least $p\,n$. Given constant $c \in (0,1)$, what is the ...
- 1,677
5
votes
0
answers
133
views
Expectation of a specific random variable on the probability space of $n\times n$ matrices over $\{0,1\}$
Let $\mathcal{G}_{n,\frac{1}{2}}$ be the probability space of $n\times n$ matrices over $\{0,1\}$ and each entry of the matrix is independently equal to 1 with probability $\frac{1}{2}$ and equal to 0 ...
- 1,997
5
votes
0
answers
327
views
Eigenvalues of Random Regular Bipartite Graphs
I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
- 161
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
4
votes
0
answers
355
views
Distribution of min/max row sum of matrix with i.i.d. uniform random variables
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $1$.
all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables ...
- 272
4
votes
0
answers
188
views
Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
3
votes
0
answers
145
views
Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
- 31
3
votes
0
answers
151
views
Largest eigenvalue divided by $n$
Let $X$ be an $n\times n$ symmetric random matrix whose diagonal is fixed as $1$, and every element in the upper triangle (excluding the diagonal) is drawn from Bernoulli($p$). The elements in the ...
- 272
3
votes
0
answers
414
views
Eigenvalue distribution of a special symmetric matrix of uniform random variables
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $0$.
all other elements in the upper triangle are uniform random variables over $[0,1]$. all ...
- 272
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
435
views
Rank of Hadamard product with random matrices
I do research in statistics and am not sure whether the following is considered research level or not in mathematics. If it isn't, I'm happy because that means the answer is probably known and I can ...
- 131
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
3
votes
0
answers
968
views
$\epsilon$-covering number of a set of rank-2 matrices
Suppose that two unit-norm vectors $\boldsymbol{a}\in \mathbb{R}^m$ and $\boldsymbol{b}\in\mathbb{R}^n$ are given with $m\leq n$. Furthermore, let $\boldsymbol{F}_{m,n}$ denote the first $m$ rows of ...
- 215
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
2
votes
0
answers
106
views
The distribution of eigenvalues of linear combinations of random unitary matrices
Suppose that $\alpha_{1},\dots,\alpha_{r}$ are non-zero complex numbers. Let $U_{1},\dots,U_{r}$ be random $n\times n$-unitary matrices. Let $A=\alpha_{1}U_{1}+\dots+\alpha_{r}U_{r}$.
I have observed ...
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
0
answers
95
views
Maximum volume submatrices of a Khatri-Rao product of matrix exponentials
My question requires quite a bit of setup, which leads to a conjecture. So I split my question into three parts, Setup, Conjecture, and Question.
Setup:
Pick any two right stochastic matrices $\...
- 237
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 ...
2
votes
0
answers
75
views
How to obtain mathematical expectation with the vector as random variable?
In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{...
- 171
2
votes
0
answers
64
views
Largest eigenvalue of two types of slightly different random matrices
Consider two types of slightly different $n \times n$ symmetric random matrices $X$. The diagonal elements of $X$ are fixed as $1$. Suppose $\frac{k}{n} \to \alpha$ for some constant $\alpha\in(0,1)$.
...
- 272
2
votes
0
answers
59
views
Min/max row-sum distribution of a symmetric matrix of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $0$.
randomly select $k$ distinct cells in the upper triangle (excluding the diagonal), and then ...
- 272
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
2
votes
0
answers
322
views
Expectation of square root of positive definite matrix
Let $U$ be a random matrix, supported on the positive-definite cone of matrices.
We denote $\sqrt{U}$ to be the principal square root of $U$. That is, the unique positive-definite matrix such that $\...
- 393
2
votes
0
answers
366
views
Convergence rate of Pearson correlation matrix
I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let $\rho(...
- 121
2
votes
0
answers
458
views
Random variable matrix exponential
I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here.
What ...
- 21
2
votes
0
answers
240
views
Radon transform and Log-concavity
This question is related to (but different from) that of Darsh Ranjan.
Is there a characterization of the functions $f:\mathbb R^n\rightarrow\mathbb R_{\ge0}$ whose Radon transform $\hat f(\omega,t)$...
- 52.3k
1
vote
0
answers
80
views
Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
- 193
1
vote
0
answers
134
views
Number of ways to place 4 kings on nxn chessboard
I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example:
In the case where the $4$...
- 47
1
vote
0
answers
81
views
Pre-positive definite functions?
A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
- 620
1
vote
0
answers
57
views
Concentration inequality for matrix martingale with dynamic upper bounds
Consider a sequence of stochastic PSD matrices $X_1, X_2, \dots, X_n \in \mathbb{R}^{d\times d}$. Let $\mathcal{F}_k = \sigma(X_1, X_2, \dots, X_{k-1})$ be the natural filtration and $Y_k = \mathbb{E}[...
- 11
1
vote
0
answers
225
views
Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
1
vote
0
answers
32
views
Probability of marking at least one row in given matrix
Let there be a matrix $\alpha=(a_{i,j})_{i\in [m], j\in [n]}$, where $a_{i,j}\in\{0,1\}$ And every row has exactly $r\le n$ ones.
We independently with probability $p$ choose some columns from this ...
- 123
1
vote
0
answers
69
views
A possible conjecture on exponential asymptotics of random recursion relations
I have come to suspect that the following is true (and have confirmed it with some numerical experiments) but I have no idea how to prove it.
Background: Let $f(z) = \sum_{n=0}^N a_n z^n$ be some ...
- 607
1
vote
0
answers
201
views
Rank of cross-covariance matrix
Let $\boldsymbol{X}=(X_1,\dots,X_p)^T$ and $\boldsymbol{Y}=(Y_1,\dots,Y_q)^T$ be two random vectors. Denote $r_x=\text{rank}(\text{Cov}(\boldsymbol{X})),r_y=\text{rank}(\text{Cov}(\boldsymbol{Y})), r_{...
- 193
1
vote
0
answers
282
views
Expected value of minimum rank of random matrices
I have $n$ random vectors ${\bf r}_i$ for $i=1,2,\dots,n$, each with dimension $1 \times m$, and $n$ random matrices ${\bf S}_i$ for $i=1,2,\dots,n$, each with dimension $M \times m$. The elements of $...
- 307
0
votes
0
answers
57
views
Class of covariance matrices invariant under permutations
I am reading a paper on covariance matrix estimation, and in this paper is introduced a class of covariance matrices:
\begin{equation}
U(q, c_0(p),M)=\{\Sigma: \sigma_{ii}\leq M,\quad \max_j\sum_{j=1}^...
- 1
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
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
82
views
The effect of channel error on the determinant of transmitted matrix
Assume the following matrix
$$
E:=\left(
\begin{array}{ccccc}
e_1 & e_2 & \cdots & e_{p-1} & e_{p}\\
e_{p+1} & e_{p+2} & \cdots & e_{2p-1} & e_{2p} \\
\...
- 313