All Questions
1,923 questions with no upvoted or accepted answers
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85
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What are good bounds on ratios of subdeterminants?
Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ?
Using the ...
- 1,893
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0
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162
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Can we give efficiently the solution of a bilinear system of equations over a finite field?
Consider a finite field $F$ and suppose we have a system of equations
$$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$
where $\alpha=(\alpha_1,...,\alpha_s)$ and $\beta=(\beta_1,.....
- 131
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123
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How to define the determinant of a morphism between graded Lie algebras?
I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...
- 1,881
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51
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Probability of non-negative matrix relaxation
Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, take $\mathscr{M}[M]=\{Q\in R_{\geq0}^{n\times n}:Q[ij]>0\iff M[ij]=1\}$.
Does $$\mathcal{P}_{n\rightarrow\infty,Q\in\mathscr{M}[M],M\in\Bbb\{0,1\}^{n\...
- 13.9k
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68
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Zero as a repeated permanental root for a matrix over a finite field
All,
Suppose $A \in Mat(n, \mathbb{F}_{q})$ for $q$ prime, $q \geq 5$, and $n \geq 2^{q-2}$ . Let $\pi_A(x)$ be the permanental polynomial for $A$. That is,
\begin{equation*}
\pi_{A}(x)=per(xI-A).
\...
- 11
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0
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484
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Classifying 1 cycle permutation matrices
Given a permutation matrix that is not full rank, is there a linear algebraic and corresponding algebraic criterion to tell if matrix contains more than one disjoint non-trivial cycle or exactly one ...
- 13.9k
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109
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Curve associated to bipartite graph
Given real biadjacency matrix $A\in\{0,1\}^{n\times n}$ of a bipartite graph with rank $r\in[2,n-1]$, denote $A(x)$ to be matrix where $0$ is replaced by $x$ and $1$ by $1-x$. Denote $$p_1(t,x)=Det(tI-...
- 13.9k
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270
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Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
- 157
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70
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Multiplicity of Ritz eigenvalues
Consider a Krylov subspace $K_m=\mathrm{span}\{v,Pv,...,P^{m-1}v\}$, for $P$ a square matrix and a nonzero vector $v$. Let $H_m$ represent the projection of $P$ (seen as an application) restricted to $...
- 333
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57
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cone and its scaled image
Let $C$ be a polyhedral cone in $\mathbb{R}^m$ defined by
$C = \{R y : y \in \mathbb{R}^m_+\}$ and $R\in\mathbb{R}^{m\times m}$.
Let $S: \mathbb{R}^m \to \mathbb{R}^m$, be a scaling map, i.e. $S = \...
- 21
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396
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Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
- 163
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109
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An exact fraction of a matrix
Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that $\...
- 6,962
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127
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Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil
Given a Hermitian indefinite pencil $(A-\lambda B)$ where both $A=A^H$ and $B=B^H \in \mathbb{C}^{n\times n}$ are possibly indefinite, it is straightforward to show that the eigenvalues are either ...
- 757
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157
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Relations in a space generated by indicator functions
Simple Question
I ran into the following seemingly simple question.
For an arbitrary set $M$ consider the real vector space generated by indicator functions $\chi_A$ of all subsets $A\subset M$. (...
- 3,513
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104
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Lower bound on difference between polynomials at moderate distance
Fix $r > 0$ and $k, n \in \mathbb{N}$. Also consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. Let $x_{1},\ldots, x_{n+1}$ be points chosen uniformly from $[-r,r]^{d}$. For $1 \leq i \...
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112
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Tools to bound the singular values of a finite sum of random matrices from below?
Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
- 111
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244
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Distribution of a signal covariance matrix
A common estimation problem in signal processing assumes the following signal model
\begin{equation}
\mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n}
\end{equation}
where $\...
- 345
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57
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Inverse of the covariance of the estimate of a covariance
I have a covariance matrix, $V_{ij}$, which (for reasons that aren't important) I'm going to call the visibilities. I have an estimator for the visibilities $\hat V_{ij}$, and I've derived that the ...
- 211
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133
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How to prove the following claim about Pseudoinverse
For real symmetric matrices $K$ and $\hat{K}$, if $u^{T}{K}u\le u^{T}\hat{K}u\le C u^{T}{K}u$ for all $u$ in the row space of $K$, then
$u^{T}{K^{+}}u\ge u^{T}\hat{K}^{+}u\ge u^{T}{K^{+}}u/C$ for all ...
- 11
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225
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Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices?
By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix $H$ its diagonal n-tuple $(H_{11},H_{22},..,H_{nn})$ for any choice of basis lies in the convex hull of the $n!...
- 1,893
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573
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Eigenvalue of a linear map over finite field
Let $ F_q $ be a finite field with $ q $ elements.
Let $ g $ be a multiplicative generator of $ F_{q^2}^* $.
It implies that
$ <g^{q+1}> = F_q^* $.
Let $ l $ be a prime greater than $ q^2-1 $...
- 503
1
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275
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Infinite matrices with a finite number of non-zero values on each row
The little bit of literature on infinite matrices I've been able to find studies a general setting in which the theory is hindered by constantly having to worry about whether or not various infinite ...
- 623
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53
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Distributing partially known data between n parties
Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ elements....
- 323
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460
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How to restructure adjacency matrix $A$ from shortest distance matrix $B$ in Network topology inference
An undirected graph with $n$ nodes could be referred to as an adjacency matrix $A$. $A=[a_{ij}]_{n×n}$ with $a_{ij}=a_{ji}=1$ standing for there being an edge between node $i$ and node $j$, and no ...
- 287
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311
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determinant inequality for symmetric positive definite matrices
Assume that A and B are symmetric, positive definite matrices of the same size.
For which set of real parameters $\alpha $ and $\beta$ the following relation holds
$\det(\frac{\alpha}{\alpha+\beta}...
- 11
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100
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Changing a nonlinear equality constraint into some conic inequality plus rank constraint
If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
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46
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the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?
Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance matrix,...
- 275
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140
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Reduce a Combinatorial problem
It is given n sets with k vectors. (k is element-wise positive or zero)
Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal.
What i also know but is ...
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1k
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Bounds for the infinity norm of the inverse for certain diagonaly dominant matrices
I m trying to analyse the stability against perturbations for a specific system of linear equations $Ax=b$.
For this, i use the standard condition number $||A||_{\infty}||A^{-1}||_{\infty}$.
Here ...
- 11
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1k
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Diagonal entries of a Cholesky factorization
Let $I$ denote an identity matrix, $E$ denote the all-one matrix of dimension $k\times k$ and $c$ some positive real number. Define $X=B(I-cE)B^T$ where $B$ is given by
$B:=\begin{pmatrix}
1 &\...
- 29
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147
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Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
- 121
1
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433
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Bounding the norm of the Dirichlet kernel as a matrix function
Consider the Dirichlet kerel:
$f(x) = 1+2\sum_{k=1}^{N}\cos(kx)$.
Now, given a diagonalizable real matrix $A$, one can consider $f(A)$, using the standard notation of matrix functions. Namely, $f(A) ...
- 73
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209
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An optimization problem on the sphere
Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector $v\in\...
- 13.9k
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105
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Dimension of $L(E,F)$
Let $E,F$ be two vector spaces over a field $k$.
$L(E,F)$ is the $k$-vector space of linear maps $E \rightarrow F$.
In ZFC, is there a functionnal relation between $dim(L(E,F))$ and $(dim(E),dim(F),|...
- 2,519
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214
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range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants
The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
- 723
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64
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A lower bound on the number of matrices whose image contains all multiples of $p^e$
Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that $(p^eR)^n\subseteq\...
- 1,305
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111
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Solve a linear equation with many variables using only 1 and -1
For a program I am writing, I would find it useful to find the least possible positive solution to a linear equation, using only -1 and 1 for roots.
For example...
...
- 111
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120
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Vanishing theorems that work in positive characteristic
Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...
- 1,981
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269
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M-matrix with nonconstant entries properties
I have a matrix $J(x)$ with $J_{ij}(x)=f_{ij}(x)$ where vector $x$ is $x=x_1, x_2, ..., x_m$. I have shown that $J(x)$ is an M-matrix for all $x$. There is known review paper by Plemmons (1977) of 40 ...
- 11
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784
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Determine lattice basis from given lattice points
I'm working on the Shortest Lattice Vector Problem (SVP) for a paper that I'm currently writing. I wish to verify whether a particular structural, namely the building block property ( refer to the ...
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205
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A generalization of strictly upper triangular matrces
Let $A = [a_{ij}]_{n\times n}$ be a real matrix with the property $a_{ij}a_{ji} = 0$. What can be said about the eigenvalues of$A$ ? I want to know when $A$ is non-singular and when $A$ is nilpotent. (...
- 191
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1k
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Do skew symmetric matrices ever naturally represent linear transformations?
I am used to thinking of matrices in terms of linear transformations, but it occurred to me that skew-symmetric matrices are a potential counterexample.
I can think of at least two examples in which ...
- 7,347
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324
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Linearization of cones
Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
- 6,962
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272
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"Stable" bounds on maximum size independent set in a graph
Suppose we have a graph $G=(V,E)$, and we want to upper bound $|I|/|V|$, where $I$ is the largest independent set in $G$. Then there is the Hoffman bound, which is $|I|/|V| \leq -\lambda_{min}/(\...
- 736
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0
answers
120
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Tensor product with $\mathbb{R}$ of an even unimodular lattice
Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$.
By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane.
Now ...
1
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0
answers
87
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Possible diagonal values of a product of matrices with some specific characteristics
Hello all,
This is a question that might or might not be related to my previous one.
Imagine you have two matrices:
Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where $M\...
- 345
1
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0
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245
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Converse of the Toeplitz-Hausdorff Theorem for the Joint Numerical Range.
Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two $N\times N$ hermitian matrices. Their Joint Numerical Range is defined as the 2-D set
\begin{align}
\mathbb{S}_2=\{[\textbf{u}^H\mathbf{A}_1\mathbf{u},\...
- 1,421
1
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0
answers
132
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Matrices with a common Fischer basis
Let $A$ be a real symmetric $n\times n$ matrix, normalized such that $Tr[A]=1$. Define a 'Fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for ...
- 31
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301
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How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?
Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now,...
- 165
1
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0
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1k
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Algebraic Independence of Polynomials in n Variables with Real Coefficients
I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...
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