# Tagged Questions

**4**

votes

**1**answer

44 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

53 views

### Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent
\begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation}
of a 3-tensor $W_{j,k,l}$ of ...

**0**

votes

**1**answer

74 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

**0**answers

62 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 ...

**8**

votes

**1**answer

123 views

### A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...

**0**

votes

**1**answer

80 views

### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...

**1**

vote

**0**answers

259 views

### how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
...

**0**

votes

**1**answer

116 views

### Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$
where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...

**7**

votes

**1**answer

776 views

### A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity:
Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...

**3**

votes

**2**answers

251 views

### On a determinant inequality of positive definite matrices

Assume that $B$ and $A$ are two positive definite matrices. Take $B^*$ a block diagonal matrix with block $B_{11}$ and $B_{22}$ of $B$. This means the following:
$$
B=\left[\begin{array}{ll}
...

**2**

votes

**0**answers

286 views

### eigenvalues of a symmetric tridiagonal matrix with zero diagonals

I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements):
$$
\left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a ...

**3**

votes

**1**answer

311 views

### What is the significance of matrix ordered algebras?

I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
...

**5**

votes

**0**answers

198 views

### On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...

**1**

vote

**0**answers

48 views

### Possible diagonal values of a product of matrices with some specific characteristics

Hello all,
This is a question that might or might not be related to my previous one.
Imagine you have two matrices:
Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where ...

**3**

votes

**1**answer

193 views

### A spectral radius inequality

Define $\rho(A)$ to be the spectral radius of a square matrix $A$. Let $S$ and $T$ be two non-negative square matrices and $h$ a real number such that $\rho(S+T) < h$. Show that $\rho((hI-S)^{-1}T) ...

**3**

votes

**2**answers

342 views

### Is Ryser's conjecture on permanent minimizers still open?

Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$.
Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if $A(k,n)$ ...

**3**

votes

**2**answers

426 views

### Average size of determinants of integer matrices?

I am interested in estimating how large determinants of matrices tend to be 'on average' given the following model: suppose we form $n \times n$ matrices $M$ such that all of the entries of $M$ are ...

**1**

vote

**0**answers

74 views

### Characterizing the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$,
$$f(x,y) = e^{\imath\pi x g(y)}$$
where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$
...

**3**

votes

**0**answers

513 views

### Matrix Transpose Similarility

A famous problem in linear algebra is that
"A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$."
I know one proof using Smith Normal form. However, I want to know an ...

**2**

votes

**2**answers

250 views

### On matrix norms

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...

**16**

votes

**1**answer

636 views

### 2x2 subdeterminants of a matrix

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...

**1**

vote

**1**answer

188 views

### What is such an equation called?

Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector?
$Ax+f(\lambda)Bx=g(\lambda)x$.

**1**

vote

**1**answer

363 views

### power of a block triangular matrix

I have a matrice in the form :
$$M =
\begin{pmatrix}
A & 0 & 0 \\\
B & A & 0 \\\
C & D & A
\end{pmatrix}
$$
where $A,B,C,D$ are diagonalizable square matrice and I want to ...

**1**

vote

**1**answer

140 views

### Scaling laws for singular values of random matrices

Assume that we have an $n\times n$ matrix ${\bf A}$ with elements drawn i.i.d. Gaussian with mean zero and variance 1.
Are there any results on the asymptotic behavior of its $i$-th largest singular ...

**3**

votes

**2**answers

683 views

### Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.

Consider a random matrix $\mathbf{A} \in \mathbb{C}^{N \times N}$ of rank $m$ with $m < N$ that follows the Wishart distribution ( http://en.wikipedia.org/wiki/Wishart_distribution ).
I have a ...

**1**

vote

**3**answers

381 views

### Number of parameters needed to specify a Hermitian matrix of rank r.

Hi,
i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.
1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank ...

**0**

votes

**3**answers

288 views

### Convex Combination of 2 hermitian matrices

Assume all the matrices I discuss about are $N \times N$. Consider any two hermitian matrices $A_1$ and $A_2$ which are indefinite. The question is, In general, for any $A_1$ and $A_2$ (both matrices ...

**2**

votes

**1**answer

241 views

### Is there a natural distance between skew hermitian matrices?

Working in machine learning, I try to find a way to compare time series, which can be considered as semi-continuous matrices belonging to $\mathbb R^{n \times \mathbb R}$ (a column corresponds to ...

**2**

votes

**2**answers

383 views

### Generalizations of Oppenheim's inequality

The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...

**1**

vote

**1**answer

177 views

### name for a matrix operation

If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?

**5**

votes

**2**answers

696 views

### A question about matrices with more details

Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda_{1},\cdots,\lambda_{r},\cdots,\lambda_{m}$ such that
...

**2**

votes

**1**answer

173 views

### Simultaneous decomposition of three projectors

A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^*PU$ and $U^*QU$ are block-diagonal ...

**5**

votes

**0**answers

198 views

### concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...

**1**

vote

**4**answers

259 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

**2**answers

201 views

### small sums of entries in submatrices - strange phenomenon

Suppose that $x \in \mathbb{R}^{n}$ is a vector of small positive fractions, i.e. $x_{i} \approx \frac{1}{n}$. The exact values are unknown. I form the matrix $M=diag(x)-\frac{xx^{T}}{2}$ which is a ...

**13**

votes

**2**answers

935 views

### Minimum off-diagonal elements of a matrix with fixed eigenvalues

Hello,
I am en engineer working in radar research. I came accross a problem I cannot seem to find math literature on it.
I can ask it in two different ways. Perhaps depending on the reader, the ...