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

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15
votes
2answers
879 views

Singular values of sequence of growing matrices

I asked this question on math.stackexchange and haven't received an answer in two weeks, so I'm repeating it here. Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \cr 1/2 & 0 ...
19
votes
1answer
806 views
+150

A curious eigenvalue inequality

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...
3
votes
1answer
65 views

Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite. $P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$, and when $A$ ...
2
votes
1answer
73 views

Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question: Bounding the absolute sum of entries of the ...
3
votes
0answers
76 views

An $\mathsf{SL(n,F)}$ decomposition problem

Given $n\times n$ square matrix $A\in\Bbb F^{n\times n}_{}$ where $\Bbb F$ is a field is there an easy way to test there is NO decomposition $A=B+C+D$ where $B,C,D\in\mathsf{SL}(n,{\Bbb F})$ are ...
1
vote
0answers
7 views

non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...
-8
votes
0answers
54 views

What are singular value of $A$? [on hold]

Let $ A = \left( {\begin{array}{*{20}{c}} {x + (\frac{3}{4} + y)i}&1&1\\ 0&{(x - \frac{5}{4}) + iy}&1\\ 0&0&{(x + \frac{3}{4}) + iy} \end{array}} \right)$, and $x,y\in ...
0
votes
0answers
61 views

Upper bound on the norm of the inverse of matrices with zero limit [on hold]

Posted here too, with no answer yet: http://math.stackexchange.com/questions/1766281/upper-bound-on-the-norm-of-the-inverse-of-matrices-with-zero-limit Let $\{L(\sigma)\}_{\sigma}$ be a family of ...
11
votes
2answers
380 views
+100

A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that $\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...
2
votes
2answers
710 views

How do you compute the dual norm of an induced norm on a subspace of a finite-dimensional $\ell^p$-normed vector space?

Say you have a finite-dimensional vector space $V$ with an $\ell^p$ norm on it. In general, the norm induced on a subspace $V_s$ of doesn't have to be another $\ell^p$ norm, so the unit sphere in ...
2
votes
1answer
205 views

Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form $$\begin{bmatrix} \pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\ A_{2n} & ...
1
vote
1answer
74 views

Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that ...
4
votes
1answer
160 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
-3
votes
1answer
89 views

Eigenvalues of cyclic tridiagonal matrix [on hold]

The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum n_i=n.$ Prove that this matrix ...
-4
votes
0answers
62 views

Eigenvalues of tridiagonal matrix [on hold]

The following matrix $T$ is result of my research on special type of balanced signed graphs of order(No. of nodes) $n$. In the matrix $T$ $n_1,n_2,...,n_k$ are positive integers such that $\sum ...
3
votes
1answer
305 views

Singular values of the sum of A and A^T

As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}(A+A^{T})$ for a stochastic $A$ (as a function of the singular values of $A$), which is generally ...
7
votes
1answer
194 views

In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
1
vote
1answer
55 views

Inverting two paraboloid relations

Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations: $$ k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right| $$ $$ ...
2
votes
1answer
169 views

Rank of a fat random matrix

Let $\mathbf{R} \in \mathbb{C}^{~n \times k} $ with $n \leq k $ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Two questions: ...
-1
votes
0answers
48 views

What will draw a shape for $L = \left\{ {\lambda \in \mathbb{C}:{s_4}(\lambda ) = {s_3}(\lambda )} \right\}$ [on hold]

Let $P(\lambda ) = \left( {\begin{array}{*{20}{c}} {{\lambda ^2} - 1} & 0 \\ 0 & {{\lambda ^2} - 2\lambda } \\ \end{array}} \right)$ and $\lambda \in \mathbb{C}$( $λ$ is a complex ...
-2
votes
0answers
23 views

Linear transformation from n-dimensinal vector space to Rn [on hold]

Suppose: U is a real n-dimensional vector space, and B = {$u_1$, $u_2$,...,$u_n$} be a basis for U, let $T: U \to R^n$ be the linear transformation defined by $$ T(u) = [u]_B $$ How to prove: ...
1
vote
1answer
73 views

Size of connected components of a graph via its spectrum

I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is: Is there a way to understand the size of each connected ...
-2
votes
0answers
82 views

proof that ${\rm SL}_n (R)=E_n(R)$ in a local ring? [closed]

I have to prove that ${\rm SL}_n (R)=E_n(R)$ and I need some help. $R =R_1\cdot R_2\cdots R_n$ , and every $R_i$ is a local ring . $E_n(R)$ is the elementary group and ${\rm SL}_n(R)$ is the special ...
1
vote
1answer
106 views

Modified interlacing of eigenvalues

Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A ...
3
votes
0answers
68 views

Number of linearly independent subsets in arbitrary set

Consider $n$-dimensional vector space $\mathbb{F}_p^n$ over finite field $\mathbb{F}_p$ and a function $a: \mathbb{F}_p^n\rightarrow\mathbb{C}$. Let $$ F(a) = \sum_{\substack{S\subset \mathbb{F}_p^n ...
27
votes
9answers
12k views

Fast Matrix Multiplication

Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in ...
2
votes
2answers
104 views

Orthogonal Procrustes problem with Absolute Values

Given a non-negative matrix $T$, how to find a orthogonal matrix $O$ minimising $$ \left\lVert \, |O| - T \right\lVert_F,$$ where $\lVert \cdot \lVert_F$ denotes the Frobenius norm, and $| \cdot |$ ...
2
votes
0answers
63 views

Eigenvalues of the imaginary part of the Symplectic action on Siegel upper half plane

Let $A,B\in M_n(\mathbb{R})$ and $U=A+iB$ unitary. $R=diag(r_1,r_2,…,r_n)$ is a diagonal matrix with $r_i>0, \forall i $. I need to calculate $\det(Ae^{-R}A^T+Be^{R}B^T)$. This matrix ...
1
vote
1answer
38 views

range of singular values of sub-matrices

Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and ...
3
votes
0answers
81 views

Can the matrix exponential be equal to the elementwise exponential [closed]

Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential ...
5
votes
4answers
776 views

Is the componentwise square-root of a positive-definite matrix also pos.-def.?

Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and $B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$. Is $B$ positive-definite whenever $A$ is? In other words: ...
1
vote
3answers
160 views

The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra

Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X) \otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ ...
-5
votes
0answers
53 views

Can we show numerically? [closed]

Ee have a decomposition of a unitary matrix $U$ by $WAW^*$ where $A$ is diagonal matrix, the symbol $^*$ means transconjugate. An infinitesimal shift $dU$ changes the matrices by $dA$ and $dW$. Can ...
10
votes
2answers
409 views

How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr) $, where $B_i$ is an arbitrary $n\times n$ ...
14
votes
1answer
498 views

Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
4
votes
1answer
10k views

vector to diagonal matrix [closed]

For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector. Is there a simple way to write this transformation ...
3
votes
1answer
184 views

Recursively calculate the determinant

A generic $k \times k$ block symmetric matrix $\Sigma$ is denoted as \begin{align} \Sigma = \begin{bmatrix}\Sigma_{11} & \Sigma_{12} & \ldots & \Sigma_{1k} \\ \Sigma_{21} & \Sigma_{22} ...
10
votes
2answers
477 views

Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative. ...
-2
votes
1answer
71 views

Maximal commutative subrings of the endomorphism ring of a vector space

Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
1
vote
1answer
437 views

Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...
1
vote
2answers
87 views

Spectral radius of a non-negative matrix after moving and replicating an element

Let $A$ be a non-negative square matrix and its spectral radius (i.e., it's largest eigenvalue) be $\rho(A)$. I need to do the following operation to $A$ and compare the resulting spectral radii. ...
1
vote
1answer
88 views

Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis. Suppose there is a matrix $$ A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] ...
3
votes
2answers
2k views

Dual operators between Hilbert spaces: with or without Riesz representation

Let $X$ and $Y$ be Hilbert spaces over the real numbers (so complex conjugation plays no role, and everything will be linear in the strict sense). Let $f : X \rightarrow Y$ be a linear continuous ...
-1
votes
0answers
18 views

Singular Vectors of Matrix sums

I have found some literature on the eigenvalue and the singular value of a matrix sum, and the relevant inequalities. However, I could not find much on the eigenvectors/singular vectors, and my level ...
1
vote
0answers
32 views

Matrix diagonalization after a completely positive transformation

I have a hermitian matrix $A$ which can be diagonalized: $$A=UDU^+,$$ where U is the unitary matrix and D is the diagonal matrix. Next, I have a completely positive transformation over it, which is ...
6
votes
2answers
1k views

Proof of a fact about traces

I'm following the open courseware content on Machine Learning from Stanford University. In the lecture notes, it is given that $$\Delta_A \ tr(ABA^TC) = CAB + C^TAB^T$$ which I tried but couldn't ...
3
votes
0answers
110 views

Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
1
vote
1answer
59 views

Spectral radius's relation with row sum

Let $A$ be a non-negative $N \times N$ square matrix with $a_{i,i}=0, 1 \leq i \leq N$. Also, let $r_i$ be the $i$-th row sum of $A$. I know that $\rho(A)$, the spectral radius of $A$, is bounded as ...
7
votes
2answers
233 views

How to estimate a specific infinite matrix sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
3
votes
2answers
71 views

Complexity of solving systems of linear diophantine equations

It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...