Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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0
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43 views

A modified notion of ranks

Given $M\in\{0,1\}^{n\times n}$ of rank $r$. Denote $p^{\Bbb Z_{\geq0}}$ to be collection of finite non-negative powers of prime $p$. Denote $\mathscr{P}_r[M,p]=\{P\in\{0,p^{\Bbb ...
0
votes
0answers
15 views

Probe permutationally matrix extreme properties-II

Call $S_{r}$, collection of $0/1$ matrices of rank atmost $r$ that increase rank if any $1$ is changed to $0$. Given $M\in\{0,1\}^{n\times n}$ of rank $r$, what is probability that $M$ could be ...
1
vote
0answers
22 views

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 ...
3
votes
1answer
66 views

Probe permutationally matrix extreme properties

Suppose given $M\in\{0,1\}^{n\times n}$ of rank $r$. Assume that changing even a single $1$ to $0$ in $M$ raises rank. Does it follow that $M$ is permutationally equivalent to a block diagonal ...
2
votes
1answer
282 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$. First, let's define two matrices: ...
1
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0answers
32 views

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 ...
0
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0answers
53 views

A combinatorial question on ranks

Denote $$\mathscr{C}[r]=\{Q\in\Bbb Z_{\geq 0,\leq 1}^{n\times n}:\mathsf{rk}(Q)= r\}.$$ $$\mathscr{D}[A,t]=\{B\in\mathscr{C}[\mathsf{rk}(A)]:\mathsf{dim}(col(A)\cap col(B))\geq t\}.$$ Given ...
-1
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0answers
59 views

Minimum rank non-negative matrix summations

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$ of rank $r$. What is minimum $k$ such that $$\mathscr{A}[b,k]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(Q)\leq k\}$$ contains $R,S$ ...
1
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0answers
38 views

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 ...
-1
votes
0answers
38 views

Orthogonal vector to arbitrary vector in R3 [on hold]

I got a vector $(0, 0, 0)^T \neq v \in R^3$. Now I want a closed formula for some orthogonal vector to $w$ (I don't care which). My problem is that if I, for instance, fix $w_1$ and $w_2$ then $w_3 = ...
2
votes
1answer
53 views

Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)

I was wondering if anybody has any suggestions on the following problem: Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...
6
votes
0answers
239 views

A matrix trace inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...
3
votes
1answer
78 views

Upper bounds on elements of a matrix

During my research I have come across matrices this type $$C=B\left(B^T B\right)^{-1}B^T\ ,$$ where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...
-1
votes
0answers
25 views

Linear combinations of columns of matrices [closed]

Suppose E is a 4 × 3 matrix with columns ⃗c1, ⃗c2, ⃗c3 and rows ⃗r1, ⃗r2, ⃗r3, ⃗r4. Let ⃗v be a 3 x 1 matrix that = [2, -1, 1] How could we express E⃗v as a linear combination of ⃗c1, ⃗c2, ⃗c3? Now ...
5
votes
0answers
138 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...
0
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0answers
95 views

Anticommuting operators with positive properties

Which classes of $M\in \mathsf M_k(\Bbb R)^{n\times n}$ admit solutions $N\in \mathsf M_k(\Bbb R)^{n\times n}$ such that $$(M\otimes N+N\otimes M)(u\otimes u)=0$$ forall $u\in \mathsf D_k(\Bbb ...
3
votes
0answers
25 views

Quasi-M matrices?

Does any body know a reference on lower triangular matrices with negative entries everywhere except for the diagonal and subdiagonal where entries are positive (when all entries are negative with ...
8
votes
3answers
268 views

Distinguishing combinatorial maps by their linearizations

Every (not-necessarily invertible) map $f$ from $[n]:=\{1,2,,,,.n\}$ to itself determines a linear map $L_f$ from ${\bf R}^n$ to itself that sends the basis vector $e_k$ to $e_{f(k)}$ for $1 \leq k ...
1
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0answers
46 views

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). ...
3
votes
1answer
95 views

A norm description for singular matrices

For $n>2$, are there norms $\parallel.\parallel_{a}$ and $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property: $A\in M_{n}(\mathbb{R})$ is singular if and only if ...
0
votes
0answers
64 views

Primitive matrix type rank range [closed]

Given a non-negative matrix $A$, we call $A$ primitive if $A^k$ has all strictly positive entries with some $k>0$. Following matrix $$\begin{pmatrix} a& b\\ c& 0 \end{pmatrix}$$ with ...
0
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0answers
12 views

Dense Symmetric Rank Deficient Linear System

What are some of the best methods for solving a Dense Rank Deficient Linear System Ax = b, where A is Dense, Symmetric but possibly Rank Deficient. I know SVD can solve it pretty nicely while ...
4
votes
1answer
64 views

On primitive type matrix ranks

Given a non-negative matrix $A$, we call $A$ primitive if $A^k$ has all strictly positive entries with some $k>0$. Given primitive $A$, is there relation between smallest $k$ such that $A^k>0$ ...
0
votes
1answer
49 views

Stationary distribution of random walk alias solving uncountably many linear equations [closed]

Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$. Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. ...
0
votes
1answer
138 views

A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function. What matrices belongs to $S$, precisely? Let ...
0
votes
0answers
53 views

Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems? $$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...
0
votes
0answers
88 views

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 ...
4
votes
1answer
134 views

Matrix-convexity of inverse of the cofactor matrix

Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ ...
1
vote
0answers
90 views

Reference: Continuity of Eigenvectors [closed]

I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, but got no answer. For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric ...
3
votes
2answers
112 views

Positive definiteness of infinite tridiagonal matrices

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix $$ A= \begin{bmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & ...
0
votes
1answer
38 views

Characterisation of a matrix ordering property

Let $n$ be a positive integer; we consider all matrices mentioned henceforth to be $n$-by-$n$ matrices. Let $A$ and $B$ be matrices wherein all entries are nonnegative (such matrices will be called ...
2
votes
2answers
148 views

Matrices with real spectrum

Assume you have a non-symmetric real square matrix of all whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix? EDIT: Is it at least similar to ...
1
vote
0answers
95 views

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 ...
0
votes
0answers
67 views

Integration over a second order tensor [migrated]

I would like to compute the mean value of a second order tensor $\mathbf{T}$ expressed in planar cylindrical coordinates. The mean value for any second order tensor is (reference [1] page 101) ...
1
vote
1answer
95 views

Matrix Submodular Inequality

Given $a,b,x > 0$ I know following the submodularity property holds: \begin{align} \frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x} \end{align} My question is, does this property ...
4
votes
2answers
197 views

Convexity of a function of matrices

Let $A$ be an $n\times n$ positive-definite matrix. Let $0<\lambda _1 \leq \lambda_2 \leq \lambda _3 \ldots \leq \lambda _n$ be the eigenvalues of $A$. Let $n\geq k\geq 1$. Is the function $f(A) = ...
3
votes
0answers
57 views

Where does this identity involving sums of Hankel-like determinants over partitions come from?

For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix ...
6
votes
1answer
219 views

Jordan decomposition of the tensor product of two matrices

I asked this question on Math.SE here, but did not get a lot of attention. I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...
1
vote
0answers
63 views

Algorithms to compute the rank of a parametrized matrix [closed]

Motivated by my question on Mathematics StackExchange and by a question by Anirbit on the same site, I ask for some references on the problem of rank computation for a parametrized matrix. References ...
8
votes
3answers
277 views

Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$

Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D ...
0
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0answers
34 views

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 ...
2
votes
1answer
241 views

Geometric mean of two matrices

Suppose $D_1$ and $D_2$ are two $3\times 3$ diagonal matrices with real positive entries on their diagonal. Let $K$ be a symmetric $3\times 3$ matrix with zeroes on its diagonal but with arbitrary ...
0
votes
0answers
36 views

Orthogonalization technique after cosparse dictionary update

I'm trying to adapt the cosparse dictionary learning (DL) approach described in Analysis K-SVD to a DL method that creates the dictionary as a union of orthonormal blocks (UONB). For this I apply the ...
3
votes
3answers
368 views

classifying space and cohomology of integer general linear group

I have obtained that the classifying space $$ BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty) $$ is the Grassmannian. I have also obtained that the mod 2 cohomology is the polynomial ...
1
vote
0answers
47 views

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 = ...
0
votes
0answers
39 views

Dual cone of a set of particular semidefinite cones

Let $X$ be a matrix variable $$X=\begin{pmatrix} x_1 & x_2 & x_3\\ x_2 & x_4 & x_5\\x_3 & x_5 & x_6\end{pmatrix},$$ define the cone as ...
0
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0answers
29 views

Smoothness of linear equation

Suppose $\beta$ is a solution to some linear equation $ M \beta = f$ where $M$ is lower triangular with all negative entries except for the diagonal and first sub-diagonal where all entries are ...
1
vote
0answers
30 views

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 ...
1
vote
0answers
72 views

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 ...
1
vote
1answer
83 views

Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric). Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix. Case $1$: $M+W\in\{0,1\}^{n\times n}$. Could ...