**-2**

votes

**0**answers

42 views

### What kind of matrix is similar to an irreducible matrix? [on hold]

For example, we know that the set of positive definite matrix is similar to an irreducible matrix. Therefore, the set of such matrices should include the positive definite matrix cone. What kind of ...

**1**

vote

**2**answers

116 views

### Norm of a matrix operator with a special structure

Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that
$$\sum_{n=1}^{\infty}\alpha_n<\infty.$$
Question: Is there any chance to evaluate the operator norm of the matrix operator
...

**1**

vote

**0**answers

62 views

### Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$.
B is any $n \times n$ n.n.d. matrix.
What will be the sharpest upper bound on the largest eigenvalue of:
...

**2**

votes

**1**answer

69 views

### Equivalence of entrywise 1-norm and Schatten-1 norm

Let $A \in \mathbb{R}^{m\times n}$ and $\|A\| = \sum_{i, j} |A_{i,j}|$.
I am looking for constants $\alpha, \beta \in \mathbb{R}$ such that
$\alpha \|A\| \leq \|A\|_* \leq \beta \|A\|$
The function ...

**2**

votes

**2**answers

198 views

### quadratic matrix equation

Find all symmetric matrices $X=X^{T}$ such that
\begin{align}
XDX^{T}=-D \quad (1)
\end{align}
where $D\ne 0$ is a real diagonal matrix.
For example, $X=iI$ satisfies $(1)$. Can you get a ...

**3**

votes

**0**answers

77 views

### Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$.
My question is the following:
Does it always exists a ...

**3**

votes

**2**answers

155 views

### What are interesting heuristics of determining how far given matrix is from a singular one?

The condition number and volume of matrix (defined as absolute value of its determinant) are things which come to mind. Is there more?
I think that over the years numerical folks (who are faced with ...

**3**

votes

**0**answers

70 views

### When is a Hankel matrix invertible?

I would like to have some conditions that render a general Hankel matrix of the form
\begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} ...

**2**

votes

**1**answer

71 views

### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...

**1**

vote

**0**answers

48 views

### Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix

$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n} $ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where ...

**9**

votes

**0**answers

300 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 ...

**3**

votes

**1**answer

92 views

### Upper bounds on elements of a matrix

During my research I have come across matrices this type
$$C=B\left(B^T B\right)^{-1}B^T\ ,$$
where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...

**4**

votes

**1**answer

144 views

### Matrix-convexity of inverse of the cofactor matrix

Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ ...

**3**

votes

**2**answers

116 views

### Positive definiteness of infinite tridiagonal matrices

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & ...

**1**

vote

**0**answers

66 views

### Algorithms to compute the rank of a parametrized matrix [closed]

Motivated by my question on Mathematics StackExchange and by a question by Anirbit on the same site, I ask for some references on the problem of rank computation for a parametrized matrix. References ...

**0**

votes

**0**answers

43 views

### Dual cone of a set of particular semidefinite cones

Let $X$ be a matrix variable
$$X=\begin{pmatrix} x_1 & x_2 & x_3\\ x_2 & x_4 & x_5\\x_3 & x_5 & x_6\end{pmatrix},$$
define the cone as
...

**1**

vote

**0**answers

66 views

### Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil

Given a Hermitian indefinite pencil $(A-\lambda B)$ where both $A=A^H$ and $B=B^H \in \mathbb{C}^{n\times n}$ are possibly indefinite, it is straightforward to show that the eigenvalues are either ...

**3**

votes

**1**answer

111 views

### Normalizing a matrix via triangular transformations

Consider the following $n\times n$ real-valued matrix:
$$
A=\begin{bmatrix}
\alpha_1 & \beta_1 & 0 &\cdots & 0 &\gamma_n\\
\gamma_1 & \alpha_2 & \beta_2 & \cdots & ...

**3**

votes

**1**answer

253 views

### How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...

**5**

votes

**1**answer

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 ...

**-1**

votes

**1**answer

91 views

### How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable?
In this context PSD means symmetric PSD.

**1**

vote

**0**answers

70 views

### Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model
\begin{equation}
\mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n}
\end{equation}
where ...

**0**

votes

**0**answers

91 views

### Question about majorization of eigenvalues after conjugation

Let $A$ and $B$ be $n \times n$ positive semidefinite matrices with eigenvalues $\alpha_1 \ge \alpha_2 \ge \ldots \ge \alpha_n$ and $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_n$ respectively. ...

**1**

vote

**0**answers

33 views

### An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...

**6**

votes

**1**answer

203 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 ...

**5**

votes

**0**answers

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 ...

**0**

votes

**0**answers

20 views

### Exact diagonalization of tridiagonal centrosymmetric matrices

It is said that one can diagonalize a tridiagonal matrix using the analytical Lanczos method http://arxiv.org/abs/cond-mat/9712283v1. In some references in it, they always say that the starting point ...

**0**

votes

**0**answers

39 views

### Random Matrix Theory: question on a quantity on page 87 in book of Bai and Silverstein 2010

Anyone else is reading carefully the book of Bai and Silverstein 2010, titled "Spectral Analysis of Large Dimensional Random Matrices"?
On page 87 of this book, when they state the final step in the ...

**2**

votes

**0**answers

149 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 ...

**7**

votes

**2**answers

490 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

**0**answers

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 - ...

**0**

votes

**3**answers

93 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

**1**answer

114 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 ...

**6**

votes

**0**answers

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.
...

**4**

votes

**2**answers

200 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

**1**answer

94 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

**0**answers

79 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

**2**answers

164 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

**2**answers

273 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

**1**answer

46 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

**0**answers

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

**0**answers

62 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

**1**answer

260 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

**1**answer

93 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

**0**answers

38 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

**2**answers

421 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**

votes

**0**answers

63 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

**2**answers

481 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

**1**answer

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 $x=(x_1,x_2,\cdots,x_n)^T$.
...

**4**

votes

**1**answer

72 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, ...