The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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19
votes
0answers
687 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
17
votes
1answer
817 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
2answers
1k 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 ...
12
votes
2answers
625 views

Does the matrix exponential preserve the positive-semi-definite ordering?

Suppose $A$, $B$, are symmetric, real valued matrices and $B-A$ is positive-semidefinite, i.e. $A≼B$. Does that imply $e^A ≼ e^B$? Would love some intuition here. I know for instance that $A≼cI \iff ...
12
votes
2answers
601 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 ...
12
votes
1answer
370 views

Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$ ...
11
votes
2answers
557 views

Multiplicative Identity for all elements in SU(n)

Let $\{P_i\}$ be a subset of $SU(n)$ such that for any $U$ in another subset (or perhaps subgroup) $H$ of $SU(n)$: $$P_1UP_2U\cdots P_mU=I$$ where $I$ is the identity element. Is there a sequence ...
11
votes
1answer
261 views

Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...
11
votes
2answers
445 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 ...
11
votes
0answers
235 views

Ratio of entries of A and log A where A is a triangular matrix

Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - ...
10
votes
2answers
422 views

Nearby matrices have nearby leading eigenvectors?

Suppose I have a symmetric positive semidefinite matrix $A$ with leading eigenvalue $1$ of multiplicity $1$ and remaining eigenvalues $\leq\epsilon$. I am told that another symmetric positive ...
9
votes
2answers
975 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, ...
9
votes
1answer
475 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
0answers
304 views

A matrix trace inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...
8
votes
1answer
282 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
2answers
495 views

About Sylvester's determinant

If $A$ is any $n \times m$ matrix and $B$ is any $m \times n$ matrix then one familiar form of the Sylvester's identity is $\det(I + AB) = \det(I + BA)$. Now somehow curiously this above identity ...
7
votes
1answer
331 views

Block Matrix determinant

Consider the $k \times k$ block matrix: $ C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \ddots & ...
7
votes
1answer
785 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
1answer
180 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
7answers
640 views

Source for roots of matrix polynomials?

A matrix polynomial is a polynomial whose variables are square $n \times n$ matrices, let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$. I am seeking a source of results on ...
6
votes
1answer
316 views

On the positivity of matrices

For given $n$, the following $n\times n$ real matrix $M=M^{T}$ is called positive, if $x^{T}M x\geq 0$ holds for all non-negative real $x_1,x_2,\cdots,x_n$, where $x=(x_1,x_2,\cdots,x_n)^T$. ...
6
votes
2answers
547 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
2answers
350 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
1answer
204 views

Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$

Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...
6
votes
3answers
293 views

Is $1/\max(i,j)$ a bounded matrix on Hilbert spaces?

I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$. It would be sufficient to know if the Lehmer matrix ...
6
votes
3answers
468 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 ...
6
votes
1answer
428 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
1answer
314 views

Is the p-norm of a matrix strictly log-convex?

Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is, $$ \|A\|_p = \sup_{x\neq ...
6
votes
1answer
337 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} ...
6
votes
0answers
185 views

Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative. ...
5
votes
4answers
3k 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 ...
5
votes
2answers
427 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
1answer
143 views

How many cospectral graphs available for a given number of nodes?

Two graphs are said to be cospectral if they have same eigenvalues wrt adjacency matrix, Normalised or Signless laplacian matrix. How many graphs has cospectral mates for a given number of nodes? We ...
5
votes
1answer
348 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
1answer
454 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
2answers
170 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}, ...
5
votes
3answers
173 views

Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e. $$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$ It is known that for arbitrary $A_{n\times n}$ $$\|T\circ ...
5
votes
2answers
270 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
1answer
159 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?
5
votes
1answer
138 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
0answers
170 views

Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
5
votes
0answers
270 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 ...
5
votes
0answers
156 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
0answers
489 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
4answers
1k 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 ...
4
votes
2answers
218 views

Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint"). I know that all the eigenvalues of matrices in this ...
4
votes
5answers
507 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
2answers
257 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
2answers
363 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
2answers
203 views

Do singular values dominate eigenvalues?

Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that $|\lambda_1| \ge |\lambda_2| \ge \cdots ...