**16**

votes

**1**answer

566 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 ...

**12**

votes

**2**answers

375 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg n.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**10**

votes

**1**answer

859 views

### Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?

Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...

**9**

votes

**1**answer

375 views

### M-matrix plus S-matrix is P-matrix?

I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two matrices and I would ...

**9**

votes

**2**answers

346 views

### Computing a large permanent

Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the matlab function written by Luke Winslow which works great for smaller matrices ...

**8**

votes

**2**answers

684 views

### On the Positive Definiteness of a Linear Combination of Matrices

In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...

**8**

votes

**1**answer

194 views

### Inequalities for Hadamard products of complex symmetric matrices

Consider a complex symmetric matrix $$ C= C_R + i C_I $$ with $C_R,C_I \in \text{Mat}_{n\times n}(\mathbb R)$ symmetric, and assume that the eigenvalues of $C_R$ are all strictly positive. Then, $C$ ...

**7**

votes

**1**answer

767 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 $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...

**7**

votes

**1**answer

130 views

### How much can I perturb a symmetric stochastic matrix and keep positive solutions?

Suppose $A$ is a symmetric stochastic $n \times n$ matrix with least eigenvalue $\lambda_n>0$. Consider real symmetric perturbations $\widehat{A}$.
How large can I take $\epsilon$ such that ...

**6**

votes

**4**answers

2k views

### Eigenvalues of infinite matrices [closed]

I am trying to find some literature on infinite matrices because I want to know how to get the eigenvalues of infinite matrices. Seriously, it seems there are very few references available. Can ...

**6**

votes

**2**answers

519 views

### Exponentiating 4 by 4 matrix analytically

Does there exist an analytical method by which i can exponentiate a 4 by 4 matrix, in the same way as the general 2 by 2 matrix case in pauli matrix basis. I have dirac matrices (which are composed of ...

**6**

votes

**2**answers

327 views

### Triangularizing a matrix with function entries

Hi Everybody!
Given a matrix, with smooth functions as arguments is there any result which say about its triangularization?
I know that, the question is in affirmative for diagonalizing a matrix ...

**6**

votes

**1**answer

236 views

### The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as
$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$
What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$?
For example, ...

**6**

votes

**1**answer

244 views

### Best rank one approximation

Assume $u,v\in\mathbb{C}^n$ are complex vectors. I was wondering if there is a closed form expression for the following problem in terms of $u$ and $v$
\begin{equation*}
\arg\min_{x\in\mathbb{C}^n} ...

**5**

votes

**2**answers

237 views

### How to check whether a matrix is completely positive or not?

The definition:
cone of completely positive matrices
$\mathcal{C}=\{\sum_{i=1}^kx_ix_i^T:x_i\in\mathbb{R}^n_+\ for \ i=1,2,...,k\}$.
I just don't knwo how to check whether a matrix belongs to ...

**5**

votes

**1**answer

281 views

### A matrix inequality involving the Hilbert-Schmidt norm

This question comes from a problem in PDEs on which I'm currently working. Let $a$ be a $3\times 3$ matrix, real symmetric and positive definite. Denote with $\|a\|^2 _ 2=\sum a_{ij}^2$ the square of ...

**5**

votes

**3**answers

289 views

### Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R ...

**5**

votes

**1**answer

418 views

### Rank of the absolute-value matrix $|M|$ vs. rank of $M$

Let $M$ be a real matrix of rank $r$ (and let us set $M=UV^T$, with $U,V^T\in\mathbb{R}^{n\times r}$, to fix the notation).
Let $|M|$ be the matrix obtained by taking the absolute value of each entry ...

**5**

votes

**2**answers

230 views

### Simultaneous maximization of two Generalized Rayleigh Ritz Ratios

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...

**5**

votes

**1**answer

91 views

### Are there any known results on numerical ranges of rank-one positive semi-definite matrices?

In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then ...

**5**

votes

**0**answers

132 views

### Energy barriers between Hadamard matrices

Hadamard matrices may be characterized as $n\times n$ real orthogonal matrices $U$ that achieve the lowest possible "energy" as defined by the (scaled and shifted) entry-wise 1-norm:
$$
E(U)=n^2 ...

**5**

votes

**0**answers

388 views

### When is the inverse diagonally dominant?

There is a large literature devoted to studying the inverses of diagonally dominant matrices. I'd like to know if there is information about a so-to-say opposite situation: we have a matrix $A$ and ...

**4**

votes

**5**answers

394 views

### Spectrum of transition matrix for symmetric random walk

I asked this question previously on math.stackexchange.com, where it had little traction.
Consider the symmetric random walk on $\{0,1,…,n\}$ with transition probabilities $P(j→j±1)=1/2$ for $0 ...

**4**

votes

**2**answers

243 views

### system of homogeneous matrix equations

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$.
One of my friend asked me the following ...

**4**

votes

**2**answers

302 views

### Optimization version of the Sylvester equation

The Sylvester equation is a matrix equation of the form $AX-XB=C,$ where $A,B,C$ are given matrices of dimension $m\times m,n\times n$ and $m\times n$ and $X$ is an unknown matrix of dimension ...

**4**

votes

**2**answers

157 views

### Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...

**4**

votes

**2**answers

149 views

### Maximal norm-1 projection

Suppose I have a real unitary matrix $U$ and a unit vector $\mathbf{x}, \|\mathbf{x}\|_2 = 1$. What is the solution to the following problem?
$$
\widehat{\mathbf{x}} = \arg\max_{\mathbf{x}, ...

**4**

votes

**1**answer

339 views

### An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...

**4**

votes

**1**answer

218 views

### Rank of a matrix with missing entries

Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows,
$
M_{A, B} = 1
$
if $A \subseteq ...

**4**

votes

**1**answer

117 views

### Epidemic threshold

Need some help / ideas to proceed. Stuck for a while on this.
In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{max}(A)$ where $\lambda_{max}(A)$ is the ...

**4**

votes

**2**answers

324 views

### Perron Frobenius with one negative pair of entries

Suppose you have a real symmetric matrix $A$ which is positive except for $a_{ij},a_{ji}$, who are negative.
While it is not generally true that the eigenvector of the dominant eigenvalue of $A$ is ...

**3**

votes

**3**answers

864 views

### The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...

**3**

votes

**2**answers

233 views

### A short question about the DFT matrix

Is the DFT matrix the unique* unitary matrix with all entries of same magnitude?
(*up to some trivial transformations)

**3**

votes

**2**answers

113 views

### a monotone relation for s-numbers

Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one ...

**3**

votes

**3**answers

672 views

### upper bounds on a certain matrix norm

Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?

**3**

votes

**2**answers

620 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 ...

**3**

votes

**2**answers

217 views

### Hadamard product and inertia

One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (i.e. entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is ...

**3**

votes

**2**answers

135 views

### Triangularizing a function matrix with smooth eigenvlaues

Given a matrix with function entries, which are smooth and homogeneous, and having smooth eigenvalues, can we find a conjugating matrix with smooth and homogeneous entries that triangularize the given ...

**3**

votes

**2**answers

521 views

### Woodbury formula

I wonder - do you know of any example where the Woodbury formula (cf. http://en.wikipedia.org/wiki/Woodbury_matrix_identity) was crucially used to prove anything?
It might be a useful computational ...

**3**

votes

**1**answer

105 views

### SDP formulation of noisy low rank matrix completion

Exact low rank matrix completion using nuclear norm minimization can be formulated as a semidefinite program (SDP). Following the notation in the paper, a convex problem for noisy matrix completion ...

**3**

votes

**1**answer

247 views

### Singular values of the sum of A and A^T

Dear all,
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 ...

**3**

votes

**1**answer

375 views

### norm of (sub)stochastic matrix

Is there any bounds for the norm of sub-stochastic matrix? (But it's not doubly stochastic matrix, I mean only the row sum is less than 1, while the column sum may not.

**3**

votes

**2**answers

629 views

### Spectral properties of the LDL^T matrix factorization

Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How ...

**3**

votes

**1**answer

176 views

### When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on ...

**3**

votes

**1**answer

75 views

### Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...

**3**

votes

**0**answers

82 views

### Infinite series of determinants

I am interested in what is known about the following class of sums. For a sequence of matrices $A_i$ (which possibly have different size), I am wondering about examples and methods for evaluating sums ...

**3**

votes

**0**answers

126 views

### Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...

**3**

votes

**0**answers

29 views

### Simultaneous Tridiagonalization of a given set of hermitian matrices?

I have a set of $N\times N$ hermitian matrices $A_i,~i=1,\dots,M$. Are there any results on the possibility of simultaneously tridiagonalizing them?

**2**

votes

**1**answer

202 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

**1**answer

533 views

### Explicit formula for Cholesky factorization in a special case

I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a ...