All Questions
614 questions
2
votes
2
answers
119
views
Correlation between the first and a random position of an ergodic bit sequence
Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
- 947
2
votes
1
answer
3k
views
Hadamard Product and Eigendecomposition
I just found this related question in here Q1.
Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary ...
- 342
2
votes
0
answers
164
views
Linear combinations of low-degree polynomials with agreement guarantee
Let $\mathbb{F}$ be a finite field of odd characteristic and let $f:\mathbb{F}_{\leq d}[x,y]\rightarrow\mathbb{Z}$ map bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$ to ...
- 125
2
votes
1
answer
248
views
How quickly can irreducible aperiodic convex combinations of permutation matrices converge to the stationary distribution?
Recall that a doubly stochastic matrix is a square matrix with non-negative entries where the sum of each row and the sum of each column is 1. The Birkhoff-von Neumann theorem states that every doubly ...
2
votes
2
answers
840
views
When can a matrix with negative entries have a completely non-negative dominant eigenvector?
Perron-Forbenius obviously answers this question for positive and for certain non-negative matrices. I want to know whether these conditions can be weakened at all. In other words, what, if anything, ...
- 31
2
votes
1
answer
325
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
- 13.4k
2
votes
2
answers
2k
views
Hessian of function of covariance matrices
Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$,
$$
\mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\...
- 125
2
votes
1
answer
385
views
Determinants of striped Hankel matrices
This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
- 43.2k
2
votes
2
answers
403
views
Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
- 23
2
votes
1
answer
242
views
Necessary conditions for existence of linear combination of these matrices to be singular
I'm doing some research in Control theory, and a stumbled with this problem. Any help is appreciated.
QUESTION
Let $P_1,\dots,P_m$ be $m$ symmetric positive definite $n\times n$ matrices with $m<n$ ...
- 344
2
votes
1
answer
1k
views
Subgradient of Minimum Eigenvalue
Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{\text{min}}$ ...
- 1,421
2
votes
1
answer
290
views
Endomorphism rings of infinitely generated free modules generated by idempotents?
Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
- 23
2
votes
0
answers
167
views
Lower bound for the sum of cosines between singular vectors of diagonally dominant matrices
Let $A \in \mathbb{R}^{n \times n}$ be a nonsymmetric diagonally dominant matrix with $a_{ij} < 0$ $\forall i \ne j$ and $a_{ii}>0$.
Let the singular value decomposition of $A$ be $A=U \Sigma V^...
- 323
2
votes
4
answers
1k
views
Change of variables in a Gaussian integral in matrix form
I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$
where ...
- 93
2
votes
0
answers
772
views
Expected mean square error of an estimation problem
Let R be an $(N+1)\times (N+1)$ Toeplitz matrix. I would be mostly interested in the case of $N\to \infty$. Let $x$ be a complex Gaussian random $N\times 1$ vector with mean zero and covariance matrix ...
- 96
2
votes
1
answer
904
views
Diagonalizability of Gaussian random matrices
Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...
- 123
2
votes
1
answer
222
views
$C=A \cdot B$ matrices and exact sequence of $DVR$-modules
I'm looking for a proof or a reference for the following statement:
Let $R$ be a DVR (Discrete Valuation Ring) and $p$ a prime element, and let $\mathfrak a$, $\mathfrak b$ and $\mathfrak c$ be ...
- 291
2
votes
1
answer
742
views
rank of a linear combination of matrices
Let $A_1,..., A_s \in M_n(\mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $\mathbb{R}$. This means that
$$
m = \min_{(c_1, ..., c_s) \in \mathbb{R}^s \backslash \{0\}}...
- 3,625
2
votes
1
answer
567
views
integral basis of orthogonal complement
Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$.
My goal is to find an ...
- 145
2
votes
1
answer
927
views
Paralel bezier curve
If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the ...
- 123
2
votes
1
answer
665
views
Covering the cone of positive semidefinite matrices by intervals
Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices?
How about a general convex cone?
For the finite case the ...
- 6,962
1
vote
0
answers
2k
views
Tensor Products and Intersections
Given two algebras $A$ and $B$, and two ideals $I, J \subseteq B$ with non-empty intersection, is it true that
$$
(A \otimes I) \cap (A \otimes J) = A \otimes (I \cap J)?
$$
(Where both sides of the ...
- 637
1
vote
0
answers
69
views
Does there exist a canonical form for normal matrices which extends the following embedding?
Given an unordered pair of complex numbers $\{w,z\}$, we can associate to it the complex matrix
$$\frac 1 2\left[\begin{matrix}w + z + \frac{\left(w - z\right)^{2} + \left|{w - z}\right|^{2}}{2 \left|{...
- 4,943
1
vote
1
answer
321
views
Solve linear system with bordered positive definite matrix
I want to solve the usual $A x = b$ system. In block form:
$$ \begin{bmatrix} B & c \\ c^{T} & 0 \end{bmatrix} \begin{bmatrix} x' \\ x_{n+1} \end{bmatrix} = \begin{bmatrix} b' \\ b_{n+1} \end{...
- 139
1
vote
1
answer
223
views
Worpitzky-like identities?
Let $$r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$$
Computations suggest that $$r_{2k}(x)=\sum_{j=0}^{(k-1)^2}a(2k,j)\binom{k^2+x-j}{k^2}$$ and $$r_{2k+1}(x)=\sum_{j=0}^{k^2-k}a(2k+1,j)\...
- 5,622
1
vote
1
answer
114
views
Is a $1_A \otimes U$ invariant subspace of $\mathcal{H}_A \otimes \mathcal{H}_B$ a product $V_A \otimes \mathcal{H}_B$?
Consider a subspace $V$ of $\mathcal{H}_A \otimes \mathcal{H}_B$, with $\mathcal{H}_A$ and $\mathcal{H}_B$ finite-dimensional Hilbert spaces, that is $1_A \otimes U$ invariant for all unitary ...
- 133
1
vote
1
answer
940
views
maximal number of mutually orthogonal vectors
Let $F$ be a field, $n$ be a positive integer. Denote by $h_{F}(n)$ the maximal dimension of a subspace $X\subset F^n$ such that $(x,y)=0$ for any two (not necessary distinct) vectors $x,y\in F^n$, ...
- 109k
1
vote
1
answer
179
views
Connection between weights in the last eigenvector (corresponding to least eigenvalue) and the corresponding column of a correlation matrix
This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated.
Consider an $n \times n$ correlation matrix A such that all the off-...
- 141
1
vote
0
answers
475
views
How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?
Let $n,d_1,\ldots,d_n > 1$ be integers, and $V_1, \ldots, V_n$ be inner product spaces over $\mathbb C$, having dimensions $d_1, \ldots, d_n$ respectively. We consider the ways in which we may ...
- 1,434
1
vote
0
answers
259
views
Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...
- 969
1
vote
1
answer
190
views
Linear independence of +/- 1 strings/vectors
Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...
- 2,288
1
vote
1
answer
1k
views
product of Gaussian random matrix and a deterministic diagonal matrix
Suppose that $G$ is an $n\times n$ Gaussian random matrix of i.i.d. entries $N(0,1/n)$ and $D$ is an $n\times n$ deterministic diagonal elements. I'd like to know if there have been results on the ...
- 640
1
vote
1
answer
171
views
transposing "unrimmed" permutations
Denote the set of $n\times n$ permutation matrices by $\mathfrak{S}_n$. The ordinary transpose preserves this group.
Given $P\in\mathfrak{S}_n$, construct the $n\times n$ matrix ${}^tP$ according to ...
- 43.2k
1
vote
1
answer
129
views
Redistribute diagonal entries of a matrix
Let $d = (d_1,...,d_k)^t$ with positive entries. Denote $D:=diag(d)$ and let $m > k$. What are sufficient conditions on $d$ and $m$ so that there exists $V \in \mathbb{R}^{m \times k}$ with:
$V$ ...
- 185
1
vote
1
answer
322
views
A particular commutator of the discrete Fourier matrix
For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
- 4,058
1
vote
0
answers
267
views
Construct special "joint SVD" from separate SVDs
Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as
$$ A = XD_AY^T \\
B = XD_BY^T $$
where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
- 61
1
vote
2
answers
1k
views
Möbius transformation by 3 points in the Minkowski model
Goal
I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images.
What I have tried
I know that a projective ...
- 534
1
vote
1
answer
367
views
Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ($\...
- 413
1
vote
1
answer
223
views
A linear algebraic q-difference equation [SOLVED]
I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and ...
- 133
1
vote
0
answers
174
views
Null Space of Parity Check Matrix
We know that if $\alpha$ be s a primitive element of $F_q$ where $q$ is a prime power then the null space of the following matrix generates a cyclic code of designed distance $\mu$[1].
$$
G_{\alpha}^{...
- 313
1
vote
0
answers
88
views
On the real and finite field rank of a $0/1$ matrix - II
Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$.
Fix a permutation ...
- 13.9k
1
vote
1
answer
153
views
Schur complement and depermuting an algorithm for $\mathsf{determinant}\bmod2$
Let $$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$ be a matrix in $\mathbb F_2^{n\times n}$ where $A\in\mathbb F_2$ and $D\in\mathbb F_2^{(n-1)\times(n-1)}$ are square.
Through the determinant ...
- 13.9k
1
vote
1
answer
207
views
Maximum number of vectors with bounds on inner products
Suppose there are 2n vectors $\{m_1,m_2,...,m_n\}$ and $\{\mu_1,\mu_2,...,\mu_n\}$. All vectors are in k-dimensional Euclid space $R^k$. The vectors satisfy:
$$m_i \mu_j \leq 0, \quad \forall i\neq j $...
- 23
1
vote
1
answer
84
views
Asymptotic property of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
- 284
1
vote
1
answer
173
views
References request: reflections in coxeter groups
Let $V$ be a vector space. A reflection is a linear map $f: V \to V$ which has an eigenvalue $1$ with multiplicity $n-1$.
Let $S_n$ be the symmetric group on $\{1,\ldots,n\}$. Then the reflections in ...
- 6,201
1
vote
2
answers
734
views
Singular matrices with integer entries
I am motivated by the following paper by Greg Martin and Erick B. Wong:
http://www.math.ubc.ca/~gerg/papers/downloads/AAIMHNIE.pdf
Here the authors prove that assuming that the entries of an $n \...
- 26.9k
1
vote
1
answer
722
views
Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets
Assume $x$ is a variable belongs to $\mathbb R \setminus \{ 0,-1,+1 \}$ and consider for all $i, j \in \mathbb N$,
$$a(i,j) = \frac{(x^{i+1} + 1)^{j-1} + (x-1)}{x}$$
then for all $n \in \mathbb N$ the ...
1
vote
1
answer
6k
views
Largest eigenvalue of the sum of Hermitian matrices [closed]
Is there an expression for the largest eigenvalue of the sum of two Hermitian matrices in terms of the spectrum of the same matrices?
- 2,099
1
vote
1
answer
201
views
Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 6,741
0
votes
1
answer
122
views
Nilpotent infinite binary matrices
Let $\text{Mat}(\mathbb{N},\{0,1\})$ be the set of all maps $A:\mathbb{N}\times\mathbb{N}\to \{0,1\}$. We define a matrix multiplication for $A, B\in \text{Mat}(\mathbb{N},\{0,1\}$) and $m,n\in\mathbb{...
- 48.9k