**0**

votes

**0**answers

51 views

### Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$.
Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is
$M[i,j]={(...

**16**

votes

**1**answer

542 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 ...

**-1**

votes

**0**answers

62 views

### When will the spectral radius of a matrix reach its minimum?

Let $A$, $B$ be two $n\times n$ matrices. How to determine $s,t\in\mathbb{N}^+$ such that the spectral radius of $A^sB^t$ will reach its minimum?

**2**

votes

**0**answers

32 views

### Why does the objectivity rule out the convexity?

In the famous work "Ball J M. Convexity conditions and existence theorems in nonlinear elasticity[J]. Archive for rational mechanics and Analysis, 1976, 63(4): 337-403", it was mentioned that the ...

**2**

votes

**1**answer

133 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. $H=\...

**0**

votes

**0**answers

35 views

### Is that possible to use stieltjes transform for multiple matrices

I have the matrix calculation with expression
\begin{equation}
\frac{1}{M}tr(\mathbf{WHH}^H\mathbf{W}^H + \mathbf{R}_{nn})^{-1}
\end{equation}
whereas $\mathbf{H} \in \mathbb{C}^{M\times K}$, $\mathbf{...

**1**

vote

**2**answers

100 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 skew-...

**2**

votes

**1**answer

71 views

### Choi type matrix condition for completely positivity on a certain operator system spanned by some unitaries

Let $B_1$ and $B_2$ be $C^*$-algebras. Let $U_1, \ldots, U_n$ be some unitaries in $B_1.$ We consider the operator system $S$ spanned by $U_iU_j^*.$
Let $\phi: S \rightarrow B_2.$
Given that the ...

**1**

vote

**0**answers

49 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 ...

**5**

votes

**2**answers

246 views

### Stabilization of the pencil of skew symmetric matrices by the orthogonal group

During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by $...

**1**

vote

**0**answers

39 views

### Matrix transformation [closed]

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 ...

**5**

votes

**2**answers

291 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:
$$
\biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\...

**0**

votes

**0**answers

89 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 \...

**0**

votes

**1**answer

62 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 \mathbb{C}^{d\...

**0**

votes

**0**answers

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 ...

**3**

votes

**2**answers

271 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{) = ...

**1**

vote

**1**answer

92 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 ...

**1**

vote

**0**answers

65 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 ...

**1**

vote

**0**answers

52 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.
...

**15**

votes

**2**answers

3k views

### Determinants in Graph Theory

In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it ...

**0**

votes

**1**answer

164 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**

votes

**1**answer

85 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 ...

**0**

votes

**1**answer

102 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**

votes

**3**answers

194 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 ...

**0**

votes

**1**answer

129 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 $n$.

**0**

votes

**1**answer

53 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 ...

**1**

vote

**0**answers

99 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 ...

**-1**

votes

**1**answer

5k views

### Derivative of log determinant and inverse

I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...

**1**

vote

**4**answers

341 views

### inverse-closed matrix spaces

Is there a known characterization of such spaces?
An example: the space of $n \times n$ matrices spanned by $I$ and $J$ (the identity and all-ones matrices, respectively) is inverse closed by the ...

**1**

vote

**0**answers

133 views

### bounds on the entries of an inverse circulant matrix

Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the ...

**1**

vote

**0**answers

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
\end{...

**4**

votes

**2**answers

172 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**

vote

**1**answer

126 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 & \\
& \...

**7**

votes

**1**answer

154 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 ...

**5**

votes

**0**answers

54 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 ...

**8**

votes

**1**answer

423 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 ...

**12**

votes

**0**answers

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 &...

**2**

votes

**1**answer

90 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

**2**answers

449 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 = ...

**0**

votes

**0**answers

29 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**

votes

**1**answer

474 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 $A_j'$ ...

**-1**

votes

**1**answer

189 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**

votes

**2**answers

256 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**

vote

**1**answer

1k views

### Reachability in graphs using adjacent matrix

Assuming a graph $G$ with $N$ nodes distributed in a $\mathcal{L}\times\mathcal{L}$ area randomly. There is an edge between two nodes if and only if the Euler distance between them is equal or less ...

**1**

vote

**0**answers

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

**1**answer

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 \mathbb{R}^...

**6**

votes

**1**answer

129 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

**1**answer

540 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?

**7**

votes

**2**answers

152 views

### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system
$$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, $...

**2**

votes

**0**answers

340 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 (...