All Questions
Tagged with matrices pr.probability
127 questions
0
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57
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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
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
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134
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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
0
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0
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66
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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
1
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0
answers
81
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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
8
votes
3
answers
595
views
Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
- 1,459
36
votes
4
answers
2k
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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
1
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1
answer
149
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Is there a necessary and sufficient condition to determine whether a number sequence can serve as the first few moments of a Radon measure?
Given a few positive numbers $(M_1, M_2,\cdots, M_K)$, they are the moments of a measure if
\begin{equation}
M_k = \int d\mu(x) x^k,\quad k = 1,2,\cdots,K.
\end{equation}
This is related to the ...
- 11
8
votes
1
answer
323
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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 $B$,
$$\...
- 128k
3
votes
0
answers
145
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Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
- 31
2
votes
2
answers
170
views
expectation and variance of the norm of a random matrix
Suppose $X \in \mathbb{R}^{n \times d}$ is a random matrix where $n > d$. Given a matrix $A \in \mathbb{R}^{n \times n}$ such that $AX$ is a zero matrix in expectation, i.e., $\mathbb{E}_{X}[AX] = ...
- 225
2
votes
1
answer
81
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Distribution of scaled Johnson-Lindenstrauss transforms
Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that
$$
\...
- 269
2
votes
1
answer
231
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Trace inverse of random PSD matrix?
Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. I am interested in the expectation of $$E_{A}(\mathrm{Tr}( (A^T A + \lambda \...
- 25
4
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1
answer
261
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How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$?
I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging.
On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{...
- 288
1
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1
answer
109
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Distribution of weight of special type of random-matrix vector product?
Let $G$ be a matrix of dimension $k \times n$ sampled uniformly randomly from $F_2^{k \times n}$. It is a well known fact that $y = xG$ is uniformly distributed in $F_2^n - \{0\}$ for all $x \in F_2^k$...
- 111
4
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0
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196
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What is the minimum nonzero rank in a random subspace of matrices?
Fix positive integers $m$, $n$, and $k\leq mn$, and draw a $k$-dimensional subspace $S\leq\mathbb{R}^{m\times n}$ uniformly from the Grassmannian.
What is known about the random variable
$R(m,n,k):=\...
- 7,633
1
vote
1
answer
252
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Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible
$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...
- 151
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
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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
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
1
vote
0
answers
57
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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
2
votes
1
answer
85
views
Judge a special positive definite matrix in probability
Assume $\mathbf{x}$ is a random vector. The question is to judge whether
$$E \{ (\mathbf{xx'})^{-1} \}- E\{(\mathbf{xx'})\}^{-1}$$
is positive definite or not.
I have no idea how to do it. Could ...
- 51
3
votes
1
answer
347
views
The covariance matrix of quadratic form, without normal assumption
Assume $\mathbf{x}$ is a random vector with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Symmetric matrices $\mathbf{A}$ and $\mathbf{B}$ are given.
Without assuming normality, how to ...
- 51
5
votes
2
answers
339
views
Existence of a specific stochastic matrix
Let $0\le x_1\le x_2\le \cdots\le x_n\le n-1$ be given. My question is as follows : Under which condition there exists a doubly stochastic matrix $M=(m_{i,j})_{1\le i,j\le n}$ s.t.
$$\sum_{j=1}^n (j-1)...
user128095
1
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0
answers
225
views
Distribution and expectation of inverse of a random Bernoulli matrix
This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
1
vote
2
answers
142
views
If $x \ge 0$ and $\mathbf{1}^Tx \le \|x\|^2$ then $\mathbf{1}^T(I - xx^T / \|x\|^2) \mathbf{1} \ge \| [\mathbf{1} - x]_+ \|^2$
Notation. Denote $\mathbf{1}=(1,1,\ldots,1)$ as the vector-of-ones in $\mathbb{R}^n$. Write the "positive part" as $[\alpha]_+ = \max\{\alpha,0\}$ for $\alpha\in\mathbb{R}$ and $[(x_1,x_2,\...
- 570
6
votes
2
answers
502
views
Shannon entropy and doubly stochastic matrices
Suppose that $A$ is a stochastic matrix. We know that if $A$ is doubly stochastic, then $H(Ap)\geq H(p)$ where $H$ is Shannon entropy and $p$ is a probability vector. Is the converse true? i.e., if $H(...
- 109
2
votes
1
answer
263
views
Limit law of eigenvalue of random matrix with mean different to 0
If $X$ denotes a $m \times n$ random matrix whose entries are independent identically distributed random variables with mean $\mu$ and $\sigma^2 < \infty$, let
$$Y = X X^T$$
with $X^T$ the ...
- 335
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
2
votes
1
answer
195
views
Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S
Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...
0
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0
answers
47
views
"Probability" for a partitioned matrix to be singular
Let $A,B\in\mathbb{R}^{n\times n}$ be two nonsingular matrices with $A\ne B$, and consider the following partitioned matrix
$$
M:=\begin{bmatrix}AA^\top + BB^\top & A^\top \Delta_1 A + B^\top \...
- 2,712
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
0
votes
2
answers
124
views
Bounding $E[\|\Sigma^{-1/2}(X-\mu)\|_2^3]$ for 2-dimensional Bernoulli
Let $X\in\{0,1\}^2$ have mean $\mu=\left[\begin{smallmatrix}p_1\\p_2\end{smallmatrix}\right]$ and $\Pr[X_1 = X_2 = 1] = p\le \min\{p_1,p_2\}$.
(Note we must have $1-p_1-p_2+p\ge 0$ for the ...
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 ...
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
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
1
vote
1
answer
684
views
Probability that random Bernoulli matrix is full rank
This is probably known already, but I could not find a quick argument.
Let $M$ be an $n\times m$ binary matrix with iid Bernoulli$(1/2)$ entries, and $n>m$. Tikhomirov recently settled that the ...
- 1,096
3
votes
1
answer
336
views
Eigenvalues of random graphs
At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
- 101
7
votes
1
answer
856
views
Trace of inverse of random positive-definite matrix in high dimension?
Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
- 2,306
2
votes
0
answers
75
views
How to obtain mathematical expectation with the vector as random variable?
In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{...
- 171
4
votes
1
answer
346
views
Rank of a random sparse matrix with nonnegative reals
I believe this should be some standard result in random matrices theory, but my initial search failed to find a definitive answer.
The question is given a random sparse matrix $M\in\mathbb{R}^{n\...
- 161
3
votes
1
answer
236
views
Mixing time and spectral gap for a special stochastic matrix
Consider the following dimension stochastic matrix,
\begin{bmatrix}
p & q & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 &...
- 103
6
votes
1
answer
1k
views
Largest eigenvalues of a (random) correlation matrix?
I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
- 380
4
votes
1
answer
372
views
Eigenvalues of random matrix conditional on positive definiteness
Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
- 505
4
votes
1
answer
3k
views
Approximating the expectation of a matrix inverse
Let
$$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$
where $A$ is a given $n \times m$ matrix (where $m \gg n$),
$$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$
...
- 67
2
votes
1
answer
136
views
Local distribution of sample covariance matrix when the number of observations/realisations is less than the matrix dimension
Given a true covariance matrix $M$ of dimension $p \times p$, we generate $n$ gaussian random vectors $X_1,..X_n \sim N(0,M)$. We then get a sample covariance matrix $M_s$ based on these $n$ ...
- 159
6
votes
2
answers
738
views
Probability of a large random integer Matrix to have zero determinant
Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$
I would like to ...
- 355
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