Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,665
questions
3
votes
1
answer
105
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 ...
4
votes
1
answer
1k
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
vote
0
answers
116
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 ...
1
vote
0
answers
51
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 $$\mathcal{P}_{n\rightarrow\infty,Q\in\mathscr{M}[M],M\in\Bbb\{0,1\}^{n\...
5
votes
1
answer
870
views
Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)
I was wondering if anybody has any suggestions on the following problem:
Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...
12
votes
1
answer
1k
views
A generalization of the Powers-Stormer inequality
The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...
3
votes
1
answer
262
views
Upper bounds on elements of a matrix
During my research I have come across matrices this type
$$C=B\left(B^T B\right)^{-1}B^T\ ,$$
where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...
7
votes
0
answers
215
views
Characterizing matrices with rank constraint
Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...
3
votes
0
answers
60
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 ...
10
votes
3
answers
481
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
vote
0
answers
67
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
1
answer
1k
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 $\...
4
votes
1
answer
137
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
1
answer
119
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. ...
1
vote
1
answer
364
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 $M=...
1
vote
0
answers
481
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 ...
6
votes
2
answers
863
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 $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ ...
2
votes
0
answers
719
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 matrix),...
3
votes
2
answers
888
views
Positive definiteness of infinite tridiagonal matrices
I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \...
1
vote
0
answers
108
views
Curve associated to bipartite graph
Given real biadjacency matrix $A\in\{0,1\}^{n\times n}$ of a bipartite graph with rank $r\in[2,n-1]$, denote $A(x)$ to be matrix where $0$ is replaced by $x$ and $1$ by $1-x$. Denote $$p_1(t,x)=Det(tI-...
0
votes
1
answer
72
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
2
answers
605
views
Matrices with real spectrum
Assume you have a non-symmetric real square matrix all of 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
0
answers
260
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 ...
1
vote
1
answer
348
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
2
answers
950
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
0
answers
130
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 $$M_f(\...
6
votes
1
answer
790
views
Jordan decomposition of the tensor product of two matrices
I asked this question on Math.SE here, but did not get a lot of attention.
I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...
1
vote
0
answers
89
views
Algorithms to compute the rank of a parametrized matrix [closed]
Motivated by my question on Mathematics StackExchange and by a question by Anirbit on the same site, I ask for some references on the problem of rank computation for a parametrized matrix. References ...
8
votes
3
answers
514
views
Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$
Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D =\...
1
vote
0
answers
67
views
Multiplicity of Ritz eigenvalues
Consider a Krylov subspace $K_m=\mathrm{span}\{v,Pv,...,P^{m-1}v\}$, for $P$ a square matrix and a nonzero vector $v$. Let $H_m$ represent the projection of $P$ (seen as an application) restricted to $...
2
votes
1
answer
449
views
Geometric mean of two matrices
Suppose $D_1$ and $D_2$ are two $3\times 3$ diagonal matrices with real positive entries on their diagonal. Let $K$ be a symmetric $3\times 3$ matrix with zeroes on its diagonal but with arbitrary ...
4
votes
3
answers
1k
views
classifying space and cohomology of integer general linear group
I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial ...
1
vote
0
answers
57
views
cone and its scaled image
Let $C$ be a polyhedral cone in $\mathbb{R}^m$ defined by
$C = \{R y : y \in \mathbb{R}^m_+\}$ and $R\in\mathbb{R}^{m\times m}$.
Let $S: \mathbb{R}^m \to \mathbb{R}^m$, be a scaling map, i.e. $S = \...
1
vote
0
answers
393
views
Bound of spectral radius of polynomial of a complex matrix
I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and $W(A)...
1
vote
0
answers
108
views
An exact fraction of a matrix
Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that $\...
3
votes
2
answers
279
views
Rank changes with matrix edits
Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).
Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.
Case $1$: $M+W\in\{0,1\}^{n\times n}$.
Could ...
0
votes
0
answers
88
views
Decomposing matrices to lower ranks
Given real matrix $M\in\{0,1\}^{n\times n}$ of rank $r$. How many $M_i\in\{0,1\}^{n\times n}$ of rank $s$ does one need to write $M'=\sum_{i=1}^ta_iM_i$ for some $a_i\in\Bbb R$ where maximum absolute ...
0
votes
0
answers
80
views
Bits of precision matrix reconstruction
We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of $\widetilde{...
1
vote
0
answers
126
views
Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil
Given a Hermitian indefinite pencil $(A-\lambda B)$ where both $A=A^H$ and $B=B^H \in \mathbb{C}^{n\times n}$ are possibly indefinite, it is straightforward to show that the eigenvalues are either ...
5
votes
1
answer
442
views
Homogeneous polynomial vector fields tangent to the unit sphere
This question has something to do with that one.
Let $n\ge1$ and $d\ge1$ be two given integers. Consider the polynomial vector fields $v=(v_1,\ldots,v_n)$ whose components $v_j$ are homogeneous of ...
3
votes
1
answer
462
views
An expectation of the product of random unitaries
I want to find the answer of
$$\int dU \ U^m X \ U^{\dagger m}$$
Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...
8
votes
2
answers
2k
views
Expectation of trace of nth power of unitary matrices
I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
0
votes
0
answers
159
views
Way to parameterise sparse multi diagonal matrix
I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$
where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is
$$
\Lambda = \begin{bmatrix}
x &...
2
votes
0
answers
86
views
Are A and A^T unitarily equivalent over a p-adic field?
Let $E/F$ be a quadratic extension of p-adic field. Let $U=\{u\in GL_2(E): uu^*=1\}$ be the unitary group of rank 2.
My question is: given a matrix $A\in GL_2(E)$ can we find $u_1,u_2\in U$ such ...
1
vote
1
answer
216
views
Probabilistic statement on matrix ranks
Given $A\in\{0,1\}^{n\times n}$ with $\operatorname{rank}(A)=r=2^{O((\log_2n)^{\frac{1}{c+1}})}$.
Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.
Does
$$\lim_{n\rightarrow\infty}\mathsf{P_{A\...
4
votes
1
answer
274
views
Can every finite abelian $p$-group with duality pairing be written as cokernel of a symmetric matrix over the $p$-adic integers?
Let $G$ be a finite abelian $p$-group (where $p$ is a prime). Suppose there exists a symmetric bilinear map $\delta\colon G\times G\to \mathbb{Q}/\mathbb{Z}$ such that the induced map $g\to\langle g,\;...
2
votes
1
answer
247
views
Phase of the inner product between the elements of an ETF
I am doing research in compressive sampling for Cognitive Radio applications. While working on a project I came across with the following question:
Is there any research about the phase of inner ...
6
votes
1
answer
653
views
Some calculus in the orthogonal group $O(n)$
How can one compute each of the following matrices, explicitly:
$$\int_{O(n)} e^{g}dg$$ or
$$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$
What is the explicite entries of the resulting ...
1
vote
0
answers
139
views
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 $A\subset M$. (...
8
votes
0
answers
391
views
Approximate singular value decomposition in Banach spaces
I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...