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

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167
votes
22answers
24k views

Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry; Is there a geometric interpretation of the trace of a matrix? This question ...
3
votes
3answers
466 views

Are the finite dimensional von Neumann algebras, singly generated?

Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then : $$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$ Question : Is it singly generated (as von Neumann algebra)? how ? ...
0
votes
0answers
49 views

Bounding the off-diagonal entries of a hermitian matrix [on hold]

I asked this question on stackexchange and didnt receive any answers, am therefore posting it here. The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = ...
0
votes
0answers
29 views

max min of ratio of quadratic forms

Consider the optimization over two vectors $x$ and $y$ $$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$ for two positive definite matrices $A$ and $B$. This problem can be ...
-7
votes
0answers
34 views

why is $\frac a 0 $ = \infty $? [on hold]

I wish to know the detailed explanation as to why $\frac a 0 = \infty$ ? I know that in physics such a fraction is meaningless but I will really appreciate mathematical point of view.
15
votes
1answer
516 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 ...
5
votes
1answer
2k views

What is the time complexity of truncated SVD

Full SVD, on an m*n matrix $A$, $[U,S,V] = svd(A)$, would cost $O(m^2n + mn^2 + n^3)$ time. But what is the time complexity if we only need the $k$ largest singular values, say, $[U_k,S_k,V_k] = ...
0
votes
0answers
70 views

Orthogonal Procrustes problem for sub-spaces?

By Orthogonal Procrustes problem I mean given matrix $A$ and $B$ finding a orthogonal matrix $R$ which most closely maps $A$ to $B$, this has a solution as shown in ...
13
votes
2answers
2k views

Left and right eigenvalues

A quaternionic matrix $A$ gives rise to a function $\mathbb{H}^n \to \mathbb{H}^n$ given by $x \mapsto A \cdot x$. This is real linear, but not complex- or quaternionic-linear (in general) if we ...
6
votes
4answers
454 views

Minimum negative eigenvalue of zero-one matrices

The following question must have been answered decades ago. For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...
1
vote
1answer
444 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 ...
7
votes
1answer
2k views

Determinant of a $4n \times 4n$ block matrix where every block is singular

I have a 4n$\times$4n matrix, which can be written as \begin{pmatrix} 0 & A &B &C \cr D& 0& E & F \cr G& H & 0 & J \cr K& L& M& 0 \end{pmatrix} each ...
9
votes
2answers
708 views

Determinant of a $k \times k$ block matrix

Consider the $k \times k$ block matrix: $$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & ...
3
votes
1answer
117 views

A question on surjectivity of a bilinear quadratic map

Let $a=(a_0, a_1, ..., a_n )$, $b=(b_0, b_1, ..., b_n )$ that belong to ${\mathbb R}^{n+1}$. Define polynomials $f_a (t)=a_0 +a_1 t+ ... + a_n t^n$ and $f_b (t)=b_0 +b_1 t+ ... + b_n t^n$ and let ...
1
vote
0answers
50 views

Showing positive stability of a matrix constructed from a positive matrix

A is a positive nonsingular matrix. Let $s>\rho(A)$. We want to show that $B\equiv\left(A^{T}A\right)^{-1}\left(sI-A^{T}\right)$ is a positive stable matrix, i.e., all eigenvalues of this matrix ...
0
votes
0answers
127 views

Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
0
votes
0answers
26 views

Lower bound of general bilinear form [closed]

Suppose I have a bilinear form $X^TAY$ where $X \in R^n, Y \in R^m$ and $Α \in R^{n \times m}$. All elements of $A$ are bounded, that is $\exists \bar a_{ij}>0:|a_{ij}|\le \bar a_{ij}, \forall ...
2
votes
2answers
87 views

Behavior of orbits under small perturbations

Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange. Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a ...
4
votes
1answer
231 views

Vector with many non-zero coordinates

Given finite field $\mathbb{F}_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least ...
1
vote
0answers
62 views

Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
4
votes
1answer
106 views

About the Eigenvalues of Orthogonal Matrix plus Perturbation

Let $O$ be an orthogonal matrix, $O^T O = I$, thus its eigenvalues lie on the unit circle, $\lambda(O)=e^{i\theta}$. Furthermore, assume the form $O = X Y$, where both matrices satisfy $X^2 = I$ and ...
0
votes
0answers
48 views

a two dimensional integer bijection

I want to try to find a specific two-dimensional linear integer bijection. This is to be used in a double sum rearrangement. It's kind of complicated, but I would really appreciate if anybody has any ...
1
vote
0answers
72 views

How to write a braiding as a matrix?

Let $V$ be the vector representation of $sl_n$. Then $V \otimes V$ is a $U_q(sl_n)$-module. Suppose that a braiding \begin{align} \Psi: V \otimes V \to V \otimes V \end{align} satisfies the ...
2
votes
1answer
125 views

Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality. As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...
4
votes
1answer
89 views

Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge ...
1
vote
1answer
331 views

Integer Polynomial solutions to functional equation

Recently I came across a functional equation which always has a polynomial with integer coefficients solution. Let $$ L_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod_{i=1}^n(x+i). $$ Problem: ...
4
votes
1answer
204 views

minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...
0
votes
1answer
70 views

Finding a vector representation for a data where we only know the inner products

I am an engineer working on speech signal processing and I have a problem that I have encountered while trying to model speech signals. The mathematical formulation is not entirely pure and I try to ...
1
vote
0answers
62 views

coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant ...
7
votes
5answers
1k views

solving Lyapunov-like equation

The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
1
vote
1answer
130 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
1answer
677 views

Nilpotent matrices related to Lie algebras of special orthogonal groups in characteristic 0

In terms of matrix theory, the question I'm led to is the following: Start with an $n$-dimensional vector space over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, which has ...
5
votes
2answers
279 views

Explicit solution to a Rayleigh quotient equation

For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method): Main equation: $$ ...
3
votes
1answer
92 views

Metabolic vs stably metabolic

Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ ...
1
vote
0answers
29 views

Orthogonal basis from corresponding points in different unknown basis [closed]

I have measurements of two points from different unknown basis (A and B). All I know is that the point in A correspondings to the point in B (by corresponds I mean they originate from the same point ...
4
votes
1answer
72 views

Perturbations on the pseudoinverse of a matrix

Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation $$ A_p = A + \Delta $$ is there a way to represent $$ (A_p)^{\star}= (A)^{\star} + f(\Delta) $$ where $(A_p)^{\star}$ ...
2
votes
2answers
228 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 ...
3
votes
0answers
84 views

Finding nearest Toeplitz matrix to a given matrix

For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change. Specifically I want find the Toeplitz ...
4
votes
1answer
92 views

Behaviour of eigenspaces of adjacency matrices after a single change to the graph

Say I know the eigenvalues and eigenvectors of an adjacency matrix of an unweighted graph. Can I say anything about the eigenvalues and eigenvectors of an adjacency matrix of a graph with one extra ...
0
votes
1answer
80 views

Eigenvectors of symmetric positive semidefinite matrices as measurable functions

I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices. I've been searching everywhere for an ...
5
votes
0answers
141 views

If $A$ is an algebra, $Sym^n(A)$ is an algebra. Where can I learn more about this algebra structure?

$\newcommand{\Vect}{\mathsf{Vect}} \newcommand{\nats}{\mathbb{N}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\Alg}{\mathsf{Alg}} \newcommand{\CAlg}{\mathsf{CAlg}} \newcommand{\Hom}{\mathrm{Hom}}$ Let ...
5
votes
1answer
285 views

Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$. The goal is to return $\widehat A, \widehat B$ such ...
15
votes
2answers
902 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
1k views

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
128 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
78 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
105 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
11 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 ...
13
votes
2answers
449 views

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
719 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 ...