The study of real and complex matrices and their algebraic and analytical properties, including: eigenvalues and eigenvectors, positive definite matrices, matrix inequalities, invariant subspaces, perturbation analysis, matrix functions.

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-2
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0answers
51 views

An analytic characterization of eigenvalues of a Hermitian matrix [on hold]

[..the following is trying to understand a certain argument of Terence Tao in a lecture notes of his..] If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. ...
1
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0answers
81 views

Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
0
votes
0answers
55 views

Hadamard perturbations to the determinant [closed]

Suppose $A$ is a matrix with nonzero determinant, and suppose $E \approxeq \mathbf{1}\mathbf{1}^T$ is close to the matrix of all ones. Is it true that the Hadamard, or entrywise, product $A \cdot E$ ...
6
votes
2answers
407 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 ...
11
votes
0answers
223 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 - ...
0
votes
3answers
75 views

Maximizing/Minimizing the Operator norms of 0-1 matrices subject to a constraint

Fix $n$ and let $B, C$ be two $n \times n$ 0-1 matrices of full rank such that $\sum_{i,j} b_{i,j}^2 = \sum_{i,j} c_{i,j}^2$, in other words they have the same number of $0$ entries and the same ...
4
votes
1answer
113 views

Counting Boolean Normal Matrices of size $2n \times 2n$

Fix $n$ a natural number. Consider the set of all $2n \times 2n$ matrices with entries from {0,1}. This is clearly a finite set. I would like to count the number of such normal matrices for fixed ...
5
votes
0answers
156 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. ...
4
votes
2answers
174 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 ...
2
votes
1answer
55 views

lower bound of a trace quadratic form [closed]

i want to find a lower bound on the following expression: $tr(AXA^T)$ in terms of $tr(X)$ where A is real $n\times n$ matrix and $X>0$ is positive symmetric. It seems that the following bound is ...
2
votes
0answers
69 views

What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
4
votes
2answers
114 views

The eigenvectors and eigenvalues of matrix geometric mean

This is a follow up question on from How to solve a non-homogeneous quadratic matrix equation?. Given the matrix $G = A(A^{-1}M)^{1/2}=A^{1/2}(A^{-1/2}MA^{-1/2})^{1/2}A^{1/2}$, where $A=-H^{-1}$, ...
3
votes
2answers
233 views

How to solve a non-homogeneous quadratic matrix equation?

I am looking to solve the following matrix equation for $G$ $$GHG + M = 0$$ where $G$, $H$, and $M$ are square, symmetric, real matrices. $H$ is negative-definite and $M$ is positive-definite. $G$ ...
1
vote
1answer
44 views

Partial Constraint of Low Rank Matrix

Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...
1
vote
0answers
72 views

Does this permanent have a closed form?

What is the closed form of this permanent? (similar to the Cauchy determinant) \begin{aligned} f(z_1,z_2,\cdots,z_N,w_1,w_2,\cdots,w_N)=\left[ \small{\begin{matrix} \frac{1}{(z_1-w_1)^2} && ...
0
votes
0answers
51 views

Eigenvalue bounds for covariance matrix

If if have a random vector $\mathbf{a}\in \mathbb{R}^n$, and I form the covariance matrix of its elements $C=\mathbb{E}[\mathbf{a}\mathbf{a}^T ]-\mathbb{E}[\mathbf{a}]\mathbb{E}[\mathbf{a}]^T$, can I ...
11
votes
1answer
201 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 ($C=\bf 0$) ...
1
vote
1answer
76 views

Connection between eigenvalues of A and its LDL decomposition

Consider an undirected graph $G$ with $N$ vertices and its adjacency matrix $n_{ij}$: $n_{ij} = 1$ if vertices $i$ and $j$ are connected by an edge and $n_{ij} = 0$ otherwise. Consider $A_{ij} \equiv ...
0
votes
0answers
33 views

Distribution of Wishart Sample Eigenvalues for Multiple Roots

I am interested in finding an asymptotic approximation to the latent roots $l_1>\dots>l_p$ of a white noise Wishart matrix $nS\sim W_p(n,I)$ as $n\rightarrow\infty$ (where $p$ is fixed). In ...
10
votes
2answers
407 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 ...
0
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0answers
48 views

Rational matrix functions theory and the state space method for the case of two variables

I know there is a well-developed theory of rational matrix functions. It can be found in a number of books by Israel Gohberg et al.: "Factorization of Matrix and Operator Functions: The State Space ...
1
vote
2answers
224 views

Taking matrix derivative with MATLAB or Wolfram Alpha [closed]

I want to take the derivate of a rather complicated matrix expression. Is it possible to do this in MATLAB or Wolfram Alpha? Here is what I am trying to calculate: \begin{equation} \frac{\partial}{ ...
6
votes
1answer
297 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 $x=(x_1,x_2,\cdots,x_n)^T$. ...
4
votes
1answer
56 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, ...
7
votes
1answer
237 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 & ...
4
votes
2answers
206 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 ...
0
votes
1answer
67 views

Efficient way to find SVD of sum of projection matrices?

Lets say that we have n matrices of data $X_i : i \in [1, n]$. All $X_i$ have the same number of rows. Their associated projection matrices are $P_i = X_i(X_i^T X_i)^{-1}X_i^T$ Also say that we have ...
0
votes
1answer
59 views

Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows \begin{eqnarray*} ||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...
3
votes
1answer
132 views

Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators. We want to look at matrices that agree in most of their entries and ...
0
votes
0answers
95 views

Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...
11
votes
2answers
535 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 ...
5
votes
3answers
159 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 ...
0
votes
1answer
76 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
0answers
129 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 ...
12
votes
2answers
520 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 ...
4
votes
1answer
61 views

An extension of the Golden Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases: $$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note that if $A=I$ then ...
19
votes
0answers
638 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 ...
5
votes
7answers
515 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 ...
2
votes
0answers
93 views

proving quasi convexity of multivariable function

Given an arbitrary $(N \times N)$ square matrix ${\bf X}$ a positive definite $(M\times M)$ matrix ${\bf T}$ a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is ...
3
votes
0answers
131 views

Method to Generate Random Mutually Orthogonal Unitary Matrices

The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
-1
votes
1answer
150 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., ...
4
votes
2answers
211 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 ...
1
vote
1answer
102 views

Matrix Generator for M/M/1 Queue Waiting Time Distribution

I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct): ${\bf W} =\left( \begin{array}{ccccc} 0 ...
10
votes
1answer
340 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 $$ ...
5
votes
1answer
271 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 ...
1
vote
0answers
72 views

Inverse of matrix of generalised harmonic numbers

For $s=0,1,\dots$ and $n=1,2,\dots$, denote $r_{n,s}=\sum_{k=1}^n k^s$. It is well-known that $r_{n,s}$ are polynomials in $n$ with leading term $\frac{1}{s+1}n^{s+1}$. Let $R_{n,s}$ be the ...
4
votes
1answer
119 views

variation of the Lieb concavity theorem

A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B: $$ (A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \} $$ for $s \in ...
0
votes
1answer
82 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, ...
5
votes
0answers
258 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 ...
2
votes
0answers
156 views

Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then $\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...