Tagged Questions

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

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10
votes
1answer
203 views

Factor a sum of products of cofactors

Let $M$ be any $n\times n$ matrix. We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$. We can write ...
7
votes
3answers
376 views

Diagonalization via the Toda flow

inAccording to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...
1
vote
0answers
99 views

Default Orientation of Vectors [closed]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...
6
votes
1answer
191 views

Horn's inequalities for n matrices

Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$ i.e. Horn's inequalities for n matrices? Can such ...
2
votes
0answers
46 views

Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...
9
votes
1answer
272 views

Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$. Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$: $$R_q = M_q^T (M_q ...
5
votes
3answers
150 views

Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e. $$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$ It is known that for arbitrary $A_{n\times n}$ $$\|T\circ ...
2
votes
2answers
102 views

Boundedness of ratio of linear functions

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
1answer
142 views

Linear map with two “incompatible” representations

Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ ...
3
votes
1answer
136 views

What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
0
votes
0answers
46 views

Uniqueness of a quadratic time-dependent matrix equation

Let $v: [0,1] \to \mathbb R^n, t \mapsto v(t)$ continuously differentiable with the property that for any constant vector $h \in \mathbb R^n$ the fact that $v(t)^{\top} h = 0$ for all $t \in [0,1]$ ...
3
votes
1answer
252 views

Number of Matrices with bounded determinant

Here's my question: Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
4
votes
3answers
217 views

Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex \begin{align} \mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\} \end{align} and consider the ...
4
votes
1answer
153 views

Isomorphism of matrix ring over ore domain

Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$? An counter example, a proof or a reference is welcomed. Thanks
6
votes
2answers
255 views

elementwise functions of positive definite matrix

The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) ...
7
votes
1answer
152 views

Finite-dimensional inverse limits of double-dual spaces

Let $k$ be a field and $\{V_i\}_{i \in I}$ a filtered projective system of $k$-spaces with transition maps $f_{ji}: V_j \rightarrow V_i$ for $i \leq j$ (for my purposes we may assume the index set is ...
1
vote
1answer
171 views

Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation: $X=c \cdot AXA' - diag(c \cdot AXA')+ I$, where (1) $A \in R^{n \times n}$ is a given matrix whose element ...
2
votes
1answer
94 views

Comparison of the smallest eigenvalues of two tridiagonal matrices

Let $n\geq2$ be an integer and $E_{ii}$ for an integer $2\leq i\leq n$ be the $n\times n$-matrix with its $ii$-entry equal to 1 and remaining entries equal zero. Furthermore, let ...
10
votes
3answers
571 views

Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...
0
votes
1answer
76 views

Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$. For fix n, we can construct n dimension linear equation ...
0
votes
0answers
100 views

Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition. What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...
0
votes
3answers
211 views

Matrix $A$ such that for all matrices $B$ the product $AB$ has a row with not a single zero

Let $A$ be a given fixed $n \times m$ matrix. We also consider matrices $B$ of dimension $m \times p$. I am interested in those matrices $A$, for which for all $B \in \mathbb R^{m \times p}$ with ...
0
votes
0answers
36 views

A Optimization problem using co-ordinates of joint numerical range.

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as \begin{align} ...
4
votes
1answer
534 views

Surprising connection between linear algebra and graph theory

What is the intuition for linear algebra being such an effective tool to resolve questions regarding graphs? For example, one can determine if a given graph is connected by computing its Laplacian ...
9
votes
2answers
189 views

A differentiable one-parameter family of codimension 2 subspaces of $\mathbb{C}^n$ cannot fill $\mathbb{C}^n$, right?

Suppose that $P(t)$ is a one-parameter family of rank 2 self-adjoint projections on $\mathbb{C}^n$ that vary analytically in the real parameter $t \in [0,1]$. I claim that there must exist a vector $x ...
2
votes
2answers
223 views

Finding the set of all $0-1$ vectors in an affine subspace

We are given a $0-1$ matrix $A$ with constant row and column sum, and we need to find out if there exists a $0-1$ vector in the solution space of $Ax = \mathbf{1}$ over $\mathbb{Q}$ (or $\mathbb{Z}$) ...
3
votes
4answers
232 views

Determinant of sum of Kronecker products

Given four real symmetric matrices $A,B \in \mathbb{R}^{n \times n}$ and $C,D \in \mathbb{R}^{m \times m}$, is there an efficient way to compute the determinant: $\det|A \otimes C + B \otimes D |$
1
vote
0answers
21 views

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 ...
1
vote
0answers
80 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space ...
0
votes
2answers
103 views

Matrix equation solving guidelines [closed]

Does anyone how to solve this matrix equation? $$ PXQ^T+P^TXQ=A$$ where all matrices are real and square. Can you provide me with some guidelines? Thanx
15
votes
1answer
267 views

A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$ I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...
1
vote
1answer
250 views

Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product?

I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$. What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product ...
0
votes
1answer
153 views

Positive Semidefinite matrix [closed]

Let $A$ be an $n\times n$ symmetrix matrix, if $\forall i$, $a_{ii}\geq |a_{ij}|,\forall j$ satisfies, can we say that $A$ is a positive semidefinite matrix? I tried to find a counter example, but ...
6
votes
1answer
200 views

On an inequality among determinants

For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite. In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$, $X\ge ...
2
votes
0answers
26 views

L is the laplacian matrix of an undirected graph, D is a diagonal matrix. What does the cofactor of L+D looks like?

We know that any cofactor of the Laplacian matrix constant, and equal to the number of spanning tree. How are the cofactors change if I just add a diagonal matrix to the Laplacian matrix. Any help ...
2
votes
0answers
110 views

An integral problem related to matrix determinant

I am stuck in an integral problem: ...
19
votes
0answers
625 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
2
votes
1answer
86 views

Space of matrices B for which there is a solution to Bx=c for a given c

Let $F$ be a field, $k$ and $m$ natural numbers with $k \leq m$, and $c \in F^m$. Is there some name for the set $\mathcal{B}_c = \{ B \in F^{m \times k}\, | \,\, \exists x \in F^k $s.t. $ Bx = c\}$ ...
3
votes
0answers
65 views

the annihilator of cokernel in a particular case

Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to ...
1
vote
1answer
180 views

Diagonalization of 4th order tensors

I have been wondering about the following problem... Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...
6
votes
1answer
259 views

Equivalence of exterior forms

Let us start with the following definition. Let $1\leqslant k\leqslant n$ and let $\omega_1,\omega_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega_1$, $\omega_2$ are equivalent, if there exists ...
5
votes
2answers
190 views

Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices

I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...
4
votes
0answers
420 views

A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
1
vote
1answer
111 views

Compressing a system of linear equations

Consider the system of linear equations $A\mathbf x=\mathbf b$ in which $A$ is an $m\times n$ matrix with $m < n$ and with the following property: Property $\Gamma$: Given $M=\{ M_1,\cdots,M_r ...
2
votes
1answer
238 views

A similar Cauchy-Schwarz inequality with linear-algebra

Let $A$ be matrix in $M_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction. Assume that $A$ and $B$ are contractions such that $I-AA^*$ and $I-BB^*$ are ...
0
votes
1answer
69 views

Reference request for: inverse of a non-singular M-matrix has all elements non-negative?

Does anyone know the best (earliest?) reference please for the proof that the inverse of a non-singular M-matrix has all elements non-negative?
0
votes
0answers
34 views

Nonlinear matrix equation (transpose) [duplicate]

Let $H$, $M$ and $N$ be 10 by 10 matrices over the integers. If $M$ and $N$ are known, how do you solve for $H$ from the following equation? $M = H N H^T$ where $H^T$ is the transpose of H.
1
vote
0answers
120 views

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 ...
2
votes
2answers
256 views

Relationship between largest eigenvalue of a positive matrix $A$ and $A∘A^T$

I'm wondering whether there is certain relationship between the largest eigenvalue of a positive matrix(every element is positive, not neccesarily positive definite) $A$, $\rho(A)$ and that of ...
0
votes
1answer
158 views

Kneser graphs eigenvalues

Basically, I want to prove that, in the Kneser graph (wikipedia has a good definition),$K_{n, m}$, if $n_{-}(A(G)) $ and $n_{+}(A(G))$ denote the number of negative and positive eigenvalues of A(G) ...