All Questions
1,925 questions with no upvoted or accepted answers
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Maximum Number of Skew-Symmetric matrices
I want to count the maximum number of rank 2 matrices in a space of certain dimension but I am stuck at some point. Any help/ suggestions are appreciated. Here is the question.
Let $\mathbb{M}_m$ be ...
- 179
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79
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Threshold rank of matrix summation
Matrices $M_1,M_2,\dots,M_{k-1},M_k\in\{0,1\}^{n\times n}$ with real ranks $r_1,r_2,\dots,r_{k-1},r_k$ respectively.
What is the rank of the matrix $M=M_1\oplus M_2\oplus\dots\oplus M_{k-1}\oplus M_k\...
- 13.9k
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101
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$XOR$ function rank of matrices $ $
Matrices $M_1,M_2\in\Bbb Z^{n\times n}$ with ranks $r_1$ and $r_2$ respectively.
What is the rank of the matrix $M=M_1\oplus M_2\in\Bbb Z^{n\times n}$ (not $\Bbb F_2^{n\times n}$) which is defined by:...
- 13.9k
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172
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A vanishing sum of symmetric matrices
Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
- 2,712
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51
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Relation between nullity of components to its parent graph
Let $G$ be an undirected graph and the corresponding adjacency matrix be $A$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ are the connected components of the induced graph $G-v$ ( the ...
- 640
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19
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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
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85
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Behaviour of the Gaussian kernel matrix at small scales
I have two questions about the behavior of the Gaussian Kernel matrix at small scales. I had asked a similar question in math.stackexchange but did not get any response. https://math.stackexchange.com/...
- 549
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123
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Siegel lemma with one contrainst
Les $A=(a_{ij})_{\substack{1\le i\le m\\1\le j\le n}}$ ($n>m$) be a matrix of integers entries. Can one determine a "small" (depending on the size of entries of $A$) solution $(x_1,x_2,\cdots,x_n)\...
- 3,998
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101
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Operator norm for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$
Suppose $C$ is a $n$ by $n$ real symmetric matrix, and $x\in R^n$. Is there an operator norm of $C$ for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$?
If I decompose $C$ into $A'A = C^{-1}$, It seems ...
- 163
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174
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Approximating symmetric matrices by symmetrized low rank matrices
Fix an integer $k$, and suppose $M$ is a real symmetric $n\times n$ matrix admitting a decomposition:
$$
M = A + A^t + B
$$
with $\mathrm{rank}(A)=k$ and:
$$
\|B\|_2 \ll \lambda_{1}(M_{|\mathrm{range}(...
- 121
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422
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Difference between largest two eigenvalues of a graph Laplacian
The difference between the smallest eigenvalue and the next-smallest of a graph Laplacian (equivalently, the difference between the largest and next-largest of the random walk Markov chain on the ...
- 1,068
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125
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smallest singular value over invertible sub-matrices
Consider the matrix $M = \begin{bmatrix} A & A B \end{bmatrix} \in R^{n \times (n+m)}$, with $A \in R^{n\times n}$, $B \in R^{n \times m}$, $m < n$, $m > 1$, $A$ symmetric positive definite.
...
- 303
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Maximum number of matrices satisfying given rank conditions
Assume that we have $2k$ matrices $S_1,\ldots,S_k$ and $\Phi_1,\ldots,\Phi_k$ over some finite field $F$ such that
(i) $S_i\in F^{l/2\times l}$ and $\dim S_i=l/2$ for any $i\in\{1,\ldots,k\}$;
(ii)...
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123
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Fullrankness of sum of time shifts
I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now.
Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
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174
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Null Space of Parity Check Matrix
We know that if $\alpha$ be s a primitive element of $F_q$ where $q$ is a prime power then the null space of the following matrix generates a cyclic code of designed distance $\mu$[1].
$$
G_{\alpha}^{...
- 313
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53
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Integral matrices with a lot of small integral vectors in the kernel
Suppose $A$ is an $m\times n$ matrix with integer coefficients. These coefficients are possibly very large, however we assume there are at least $K_C$ vectors $x\in\mathbb Z^n$ with $\max_i |x_i|\leq ...
- 131
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188
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Maximum singular value of sum of an Hermitian and an anti-Hermitian matrix
Let $H$ be an $n\times n$ Hermitian matrix and $A$ an $n\times n$ anti-Hermitian matrix, i.e. $H^\dagger = H$, $A^\dagger = -A$. Consider their sum $S= H+A$. Let $\{\sigma_i(S)\}_{i=1,\dots,r}$ denote ...
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131
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A mixed Vandermonde-Wronskian matrix
I am trying to prove that a matrix of the following form is generically nonsingular:
$A= \begin{bmatrix}
1&1&1&1 \\
f_1 & f_2 & f_3 &f_4 \\
(f_1 -\frac{d}{dt}).f_1& (...
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96
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Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?
Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring.
I want to prove
that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
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43
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Is $L'L_\text{in}+L_\text{in}'L$ positive semi-definite?
Assume that $A$ is the adjacency matrix of a strongly connected directed graph, that is, $A$ is non-negative and irreducible. Let $$L_\text{in}=D_\text{in}-A',\;L=D_\text{in}-A'+D_\text{out}-A$$ where ...
- 11
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110
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Tail bound without independence
Suppose $X_i , X_j\in \mathbb{R}^d$ are gaussian vectors and $A$ is an $n\times n$ symmetric PSD matrix where $A_{ij} = f(\|X_i-X_j\|_2), \quad i,j\in 1,\ldots,n\;$ for some non-negative Lipschitz ...
- 195
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291
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Outer product of Gaussian vectors conditioned on their scaled sum being large
Consider $e\in \mathbb{R}^d$, for which $e_1, \ldots, e_d$ are independently drawn from a Gaussian, $e_i \sim \mathcal{N}(0, \epsilon)$. Let $\mu \in \mathbb{R}^d, ||\mu||_2 \leq D$. Then, what is ...
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56
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A topology for which symplectic forms are dense in skew forms
Let $V$ be a vector space over an algebraically closed field. Let $S$ denote the vector space of skew-symmetric bilinear forms on $V$. When $V$ is finite dimensional the subset of $S$ consisting of ...
- 599
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266
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Interlacing Suleĭmanova spectra
A set of real numbers $\{\lambda_1, \dots, \lambda_n \}$, $n \geq 1$, is called a Suleĭmanova spectrum if it contains exactly one positive value and $\sum_{i=1}^n \lambda_i \geq 0$. (It is well-known ...
- 1,719
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128
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How to efficiently evaluate the action form of a matrix power?
Binary exponentiation is a well-known method for evaluating positive integer powers of a matrix, $A^p, \; A\in\mathbb C^{n\times n},\,p\in\mathbb Z^+$.
However, I am not seeing an obvious way to ...
- 19
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348
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rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
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283
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Strict monotonicity of conditional variances
Let $K \geq 2$ be a positive integer and $C$ be any $K \times K$ non-singular matrix (if necessary, can assume that all $K$ rows of $C$ are needed to span the coordinate row vector $e_1'$). For ...
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When is $F(X)BF(X)$ operator monotone, if $F(X)$ is operator monotone?
Let $\Omega_{n}$ denote the cone of $n\times n$ real symmetric positive definite matrices, and consider $F:\Omega_{n} \mapsto \Omega_{n}$. For $X,Y \in \Omega_{n}$, the matrix valued function $F(\cdot)...
- 1,538
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An analogue of Hermitian matrix - does it exist?
Let $k$ be any field and $R\subseteq M_s(k)$ be a subring of $s\times s$ matrices over $k$. Identify $k$ with the scalar matrices, so that $k\subseteq R$. Let $A\in M_n(R)$ be an $n\times n$ matrix.
...
- 3,041
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114
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Intersecting vector spaces defined over different fields
Let $K_1, K_2$ be subfields of $K$, let $k = K_1 \bigcap K_2$, let $V_1$ be a $K_1$-vector space, $V_2$ be a $K_2$-vector space, both of them subsets of a $K$-vector space $V$.
How can I compute a $k$...
- 314
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208
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Additive perturbation bounds on the eigenvectors of a Hermitian matrix
I am reading this paper:
http://society.math.ntu.edu.tw/~journal/tjm/V16N1/TJM-258.pdf
where the authors find additive perturbation bounds on the matrix of the eigenvectors of a Hermitian matrix. I ...
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Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
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187
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A question concerning positive definite matrix functions
Let $C(e^{i\theta})$ be an $m\times n$ ($m\ge n$) matrix-valued continuous function of $\theta\in[-\pi,\pi]$. Let $A_1(e^{i\theta})$ and $A_2(e^{i\theta})$ be two $n\times n$ positive definite matrix-...
- 2,712
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Which weighted directed hypergraphs have incidence matrix of full rank?
what are the necessary conditions for a weighted directed hypergraph to have an incidence matrix of full rank?
In this context, we can define the incidence matrix as follows:
Let $V = \{v_1,v_2,...,...
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125
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Estimating $\ell^p$ and $\ell^q$ norms on a convex cone
For $1 \le p \le q \le \infty$, I need an inequality bounding the $\ell_q$ norm from above by the $\ell_p$ norm on $\mathbb{R}^n$: finding a $\lambda$ so that
$$
\Vert v \Vert_q \le \lambda \Vert v \...
- 10.1k
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228
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Maximum number of mutually orthogonal $n$-bit sequences
What is the maximum number of mutually orthogonal $n$-bit sequences can we construct? And how to construct them? A trivial example is using the Hadamard matrix, but we can only build $n$ orthogonal $n$...
- 367
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A Procrustes problem in spectral norm
For matrices $X,Y \in \mathbb{R}^{m \times r},m>r$, let
$$d(X,Y) = \min_{R \in \mathbb{R}^{r \times r}, \, R^\top R=I} \|X-YR\|_2^2$$
Given $Y$, if the $r$-th singular value of $Y$ satisfies $\...
- 85
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128
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Low-rank approximation of sub-sampled matrix
Considering a large data matrix $X$ with zero-centered columns that is assumed to be approximately low-rank, it is common to do PCA and project the data onto the top few principal components, and use ...
- 111
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209
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Finding the tensor product form answers of a quadratic equation
Let $\mathbf{x}_1$ and $\mathbf{x}_2$ be column vectors with entries belonging to the set $\{0,1\}$. In addition, let $\mathbf{A}$ be the adjacency matrix of an arbitrary graph. I want to find vectors ...
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286
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Generalized eigenvalue problem with nonnegative eigenvector constraint
Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}...
- 11
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75
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Collections of vectors with a small scalar product
Consider $m$ unit vectors in $\mathbb{R}^n$ with positive coordinates so that $m>n$. How does one pick the $m$ unit vectors so that the largest inner product of them is the smallest?
- 2,288
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483
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minimize norm of matrix product
I have the matrix Product $PAP^H$ and I need to minimize $\|(PAP^H)^{-1}\|^2$ (over $P$ and Frobenius norm).
$A$ is a positive definite Hermitian matrix and $P$ has the structure
$$P=\left[\begin{...
- 129
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149
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Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
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460
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Orthonormal basis of matrices
I am asking if somebody knows how to do or is aware of the following construction:
Let $n \in \mathbb{N}$ be given, then take an arbitrary matrix $A \in \mathbb{C}^{n \times n}$. Then, there are maps ...
- 21
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127
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What is the analogue of expansive matrix for automorphisms?
We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
- 41
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71
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Name for a Specific Planar Linear Transformation
Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular, with the biggest circle through those points and, how can they be determined ...
- 13.2k
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481
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psd condition for matrix completion
The nuclear norm minimization for the matrix completion problem is given by
\begin{align}
\textrm{minimize } \quad &\|X\|_{*}\\
\textrm{subject to } \quad & X_{ij}=M_{ij} \quad \forall (i,j)...
- 221
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79
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Evaluating a hopeless algorithm for recovering sparse solutions to $Ax=b$ over a finite field
Given a matrix $A \in F^{n \times m},$ $m>n,$ with rank $n$ and in row-reduced echelon form and a nonzero vector $b \in F^n$: (1) Perform elementary row operations to obtain $b_1 \not = 0$ and $b_k=...
- 259
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Analytic formula for minimizing the maximum inner product of a set of vectors
Given $x_j\in\mathbb{R}^n$, $j=1,\ldots,p$, find
$$
\widehat{w} \in \arg\min_{\Vert w\Vert=1}\max_{1\le j\le p} |\langle w,x_j\rangle|.
$$
I am also interested in the special case where we further ...
- 710
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73
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Is there any notion of a slack-matrix (and its rank) for a hyperplane arrangement?
I am assuming that the common lineality space of the hyperplane arrangement has already been factored out. So we are looking at an ``effective" arrangement in lower dimensions where all the polyhedra ...
- 2,246