Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to [tag:linear-algebra]). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur ...

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38 views

Matrix transformation [on hold]

I want to show that $(I-H^T(-s)H(s))^{-1}$ has no poles on the imaginary axis with $H(s)=C(sI-A)^{-1}B$ and $H^T(-s)=-B^T(sI+A)^{-T}C^T$ is equivalent to $M_\gamma$ has no purely imaginary ...
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39 views

Number of Skew Symmetric Matrices of fixed rank

The number of symmetric matrices of order $n$ and rank $r$ over finite fields has been counted e.g. http://www.math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf Is the number of ...
5
votes
2answers
276 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: $$ ...
0
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1answer
54 views

Restricted Isometry Property for Discrete Fourier Transform Matrix

I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in ...
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0answers
37 views

Gramian of a permutation group orbit

let $W\in R^{d\times k},d>k$. Suppose that the associated gramian has the following structure: $$ W^TW=(P_{1}t,\cdots,P_{k}t) $$ with the set $\{P_{i},i=1,\cdots,k\}$ forming a group of ...
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0answers
88 views

Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers. Does $$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in ...
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1answer
90 views

Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists. Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory). Preferably the ...
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0answers
52 views

The state-transition-matrix of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
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0answers
51 views

inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation. ...
0
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1answer
157 views

$P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable. $Z$ is eigenvalue of $P(Z )$ if ...
5
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1answer
82 views

Optimization of a function of a positive definite matrix and its inverse

This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved. Suppose I have two real, positive ...
3
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2answers
264 views

A question on determinant of a matrix polynomial

Let ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$. ${\rm{P(}}\lambda {\rm{) = ...
0
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1answer
97 views

Fixed point of quantum operations

A quantum operation is defined as \begin{equation} \varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger} \end{equation} where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...
7
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3answers
189 views

Is there a standard name for the following type of linear operator?

Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...
1
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0answers
96 views

Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
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0answers
38 views

Limits on a parameter $\alpha$ to get positive definite matrix

Given positive semi-definite $n\times n$ matrices $B_k$, how would I go about getting the limits on $\alpha_k$ such that the expression \begin{equation} \mathbb{I}-\sum_{k=1}^{m-1}\alpha_kB_k ...
0
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1answer
124 views

Matrix inequality between a traceless matrix and identity

Given a traceless matrix $C\in M_n(\mathbb{F})$, i.e., tr$(C)=0$, what is the relationship between tr$|\mathbb{I}+C|$ and tr$|C|$? The two matrices are of dimension $n$.
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1answer
50 views

Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form $A = P^TLDL^TP$, where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
4
votes
2answers
160 views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
1
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1answer
114 views

Determinant of discrete Laplacian

It can easy be shown by induction that the determinant of the $(N-1)\times (N-1)$ matrix $$\begin{pmatrix} 2 & -1 & & \\ -1 & 2 & \ddots & \\ & ...
5
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53 views

convex hull of all-ones principal submatrices

For a subset $S$ of $\{1,\ldots,n\}$, let $\mathbf{1}_S\in\{0,1\}^n$ denote the indicator vector of $S$, with a $1$ on the $i$th coordinate iff $i\in S$. Let $\mathcal{X}$ denote the convex-hull of ...
7
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1answer
152 views

approximate stationary distributions of a doubly stochastic matrix and its supports

Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...
2
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1answer
86 views

How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite?

Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives ...
6
votes
2answers
392 views

Parametrization of positive semidefinite matrices

We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition: $$ A = ...
12
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0answers
187 views

How does a permutation $P$ affect the singular value $\sigma_{\text{max}}(Q^\top P^\top Q)$ for orthogonal $Q$?

Let $q_i$ for $i=1,\ldots,m$ be the columns of the matrix $Q\in\mathbb{R}^{n\times m}$, $n\geq 2m$, which are pairwise orthonormal ( i.e. $q_i^\top q_j = \begin{cases} 1 & \text{if}\quad i=j \\ 0 ...
0
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0answers
28 views

Reducing the degrees of freedom in unitary columns

Let $U = diag([U_1, U_2, ..., U_N])$ be a block-diagonal $NM \times NM$ unitary matrix, where each $U_j$ is a unitary $M \times M$ matrix. Furthermore, let $Q_e = kron(I_M, Q)$ be the Kronecker ...
17
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1answer
452 views

Almost commuting unitary matrices

Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
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1answer
184 views

How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]

(Sorry, I do hate editing this many many times but let me try the last time) Gelfand's formula says that $$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$ I am wondering whether there is any ...
7
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2answers
253 views

What is the most efficient way to factor a matrix into a given set of generators?

I am studying finite index subgroups of certain finitely presented groups. The particular conditions on my groups make this problem easier than I phrase it here, but I am curious about a more general ...
1
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0answers
76 views

Eigenvalues of a partitioned self-adjoint matrix

This is a repost of the same question on MSE (with no reply/comment): http://math.stackexchange.com/questions/1454314/eigenvalues-of-a-partitioned-self-adjoint-matrix I would be grateful just for a ...
2
votes
1answer
59 views

Regularity of decomposition of matrix-valued function

Here is the problem. Suppose I have a positive definite matrix-valued function $A\colon \mathbb{R}^n\to \mathbb{R}^{n\times n}$. Then we know that there is a matrix-valued function $B\colon ...
6
votes
1answer
120 views

“Additive version” of Kronecker product

Let $A$ and $B$ be two square matrices with complex entries. Let $\lambda_1, \ldots, ,\lambda_n$ be the Eigenvalues of $A$ and $\mu_1, \ldots, ,\mu_m$ be the Eigenvalues of $B$. Then the Eigenvalues ...
15
votes
1answer
527 views

Is SL(n,Z[x]) generated by transvections?

Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?
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63 views

$l_{\infty}$ norms of matrix perturbations

Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension. What needs to be the bounds on (which?) norm of $B$ to ensure that ...
5
votes
1answer
83 views

when does elementwise-log preserve positive-semidefiniteness?

Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X_{ij}=\log(1+Z_{ij})$. Are there some simple sufficient ...
2
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0answers
337 views

Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way. All the ${\lambda}_i$ are distributed the same way with chi-square ...
9
votes
3answers
492 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
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. ...
6
votes
2answers
158 views

Existence and characterization of transitive matrices?

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following: For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw ...
4
votes
1answer
109 views

almost diagonal Positive semidefinite Matrix

Consider the set $\mathcal{D}_n$ of $n$-dimensional positive semidefinite matrices. A matrix $M\in \mathcal{D}_n$ is called $\epsilon$ diagonal in trace distance if there is a diagonal matrix $D\in ...
3
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2answers
140 views

generalization of result on K_1 of $SL(n,R)$

Let R be a "nice" ring with 1 (e.g. Euclidean domain). Then the subgroup E(n,R) generated by the elements $I+te_{i,j}$ is equal to $SL(n,R)$. My question is as follows: Instead of $SL(n,R)$ I look ...
1
vote
1answer
44 views

rank minimization over vector subsets

Let $S$ be a set of $n$ vectors from $\mathbb{Q}^d$. For every $k=1,2,\dots,n$, define $$r_k = \min_{T\subset S, |T|=k} \mathrm{rank}(T),$$ where $\mathrm{rank}(T)$ is the rank of a matrix formed by ...
36
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4answers
2k views

The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture. (The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
8
votes
2answers
466 views

On closest unitary matrix

In this question $\|A\|_p$ is the normalized $p$-th Schatten norm which is defined to be $\left(\mathbb E_{i} \lambda_i^p\right)^{1/p}$, where $\lambda_i$ are singular values of matrix $A$. Suppose ...
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 ...
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0answers
90 views

Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$. Define a sequence of banded ...
8
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1answer
419 views

Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
9
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0answers
97 views

What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$. When exactly are two unitary matrices related in this ...
1
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0answers
139 views

Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$. B is any $n \times n$ n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: ...
2
votes
2answers
242 views

quadratic matrix equation

Find all symmetric matrices $X=X^{T}$ such that \begin{align} XDX^{T}=-D \quad (1) \end{align} where $D\ne 0$ is a real diagonal matrix. For example, $X=iI$ satisfies $(1)$. Can you get a ...