**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 $\left(B^T B\right)^{-1}$ ...

**5**

votes

**0**answers

144 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...

**4**

votes

**1**answer

118 views

### Non-negative decomposition of a non-negative matrix

Consider a matrix $A\in{\bf M}_{n\times m}({\mathbb R})$, whose entries are non-negative. Let $r$ be the rank of $A$.
It is well-known that $A$ decomposes as $x_1y_1^T+\cdots+x_ry_r^T$ with ...

**3**

votes

**1**answer

97 views

### A norm description for singular matrices

For $n>2$, are there norms $\parallel.\parallel_{a}$ and $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property:
$A\in M_{n}(\mathbb{R})$ is singular if and only if ...

**2**

votes

**2**answers

176 views

### A line bundle over the manifold of singular matrices

According to answer of Denis Serre to this question, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$
So we ...

**1**

vote

**1**answer

151 views

### A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
What matrices belongs to $S$, precisely?
Let ...

**1**

vote

**0**answers

82 views

### Spectrum of primitive nonnegative integer matrices

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$.
Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...

**2**

votes

**2**answers

152 views

### Matrices with real spectrum

Assume you have a non-symmetric real square matrix of all whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix?
EDIT: Is it at least similar to ...

**1**

vote

**0**answers

106 views

### Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...

**0**

votes

**0**answers

111 views

### A quantity associated with an algebraic variete

Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.
Is there a geometric or algebra geometric interpretation for the following quantity:
The maximum number $k$ such that ...

**1**

vote

**1**answer

110 views

### Matrix Submodular Inequality

Given $a,b,x > 0$ I know following the submodularity property holds:
\begin{align}
\frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x}
\end{align}
My question is, does this property ...

**11**

votes

**0**answers

251 views

### Why is a matrix pencil called a pencil?

I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on.
I am aware that even Gantmacher 1959 has this terminology however I don't know ...

**6**

votes

**1**answer

241 views

### Jordan decomposition of the tensor product of two matrices

I asked this question on Math.SE here, but did not get a lot of attention.
I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...

**8**

votes

**1**answer

137 views

### Is the center of the automorphism group of a von Neumann algebra M trivial whenever M is a factor?

Question: Is the center of the automorphism group of a von Neumann algebra $\mathscr{M}$ trivial (=$\{\mathrm{id}\}$) whenever $\mathscr{M}$ is a factor (=$\mathscr{M}$ has center $\{\lambda I; ...

**3**

votes

**1**answer

222 views

### Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...

**0**

votes

**0**answers

41 views

### Orthogonalization technique after cosparse dictionary update

I'm trying to adapt the cosparse dictionary learning (DL) approach described in Analysis K-SVD to a DL method that creates the dictionary as a union of orthonormal blocks (UONB).
For this I apply the ...

**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

38 views

### Bound of spectral radius of polynomial of a complex matrix

I am trying to prove or disprove the following inequality.
$$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$
where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and ...

**1**

vote

**0**answers

75 views

### An exact fraction of a matrix

Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that ...

**2**

votes

**1**answer

123 views

### Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).
Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.
Case $1$: $M+W\in\{0,1\}^{n\times n}$.
Could ...

**0**

votes

**0**answers

43 views

### Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of ...

**0**

votes

**0**answers

41 views

### Way to parameterise sparse multi diagonal matrix

I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$
where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is
$$
\Lambda = \begin{bmatrix}
x ...

**3**

votes

**0**answers

89 views

### Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...

**4**

votes

**0**answers

81 views

### Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries).
Can we partition this union into at most $n$ rectangles?
I think it's pretty ...

**3**

votes

**1**answer

119 views

### Is there a standard notation for off-diagonal transpose?

Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$,
its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$.
But is there a conventional way of notating the matrix
...

**5**

votes

**1**answer

426 views

### Some calculus in the orthogonal group $O(n)$

How can one compute each of the following matrices, explicitly:
$$\int_{O(n)} e^{g}dg$$ or
$$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$
What is the explicite entries of the resulting ...

**1**

vote

**0**answers

187 views

### A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...

**4**

votes

**2**answers

207 views

### An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
the first one is the adjoint ...

**3**

votes

**1**answer

214 views

### Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...

**1**

vote

**0**answers

128 views

### Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...

**6**

votes

**1**answer

231 views

### Subadditivity of the square root for matrices

For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices?
If not, ...

**0**

votes

**0**answers

45 views

### About bi-stochastic and symmetric matrix

If you have an bi-stochastisc and symmetric matrix, what you can say about the second largest eigenvalue of this matrix? To be more precise, I would like to find upper bounds for it looking for the ...

**1**

vote

**1**answer

60 views

### Are certain normal matrices circulant? (Part 2)

Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $
A^TA=\begin{pmatrix}
a & b \\
b & \ddots
\end{pmatrix}$, for $b>0$.
As a user observed in the solution of Part 1 ...

**0**

votes

**2**answers

95 views

### Simple Spectrum of Jacobi matrices

I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...

**5**

votes

**0**answers

66 views

### Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...

**0**

votes

**1**answer

143 views

### Are these particular kinds of matrices well known?

Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are ...

**0**

votes

**2**answers

98 views

### About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...

**1**

vote

**0**answers

42 views

### Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...

**6**

votes

**2**answers

135 views

### Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that
$$
A^TA=\begin{pmatrix}
a & b & \cdots & b\\
b & a & \ddots & \vdots\\
...

**1**

vote

**1**answer

76 views

### QR decomposition of matrix [closed]

I have matrix $M = \begin{pmatrix} A & B \\ B^T & 0\end{pmatrix}$, where $A$ is $N\times N$, $B$ is $N\times 2$ and I have $Q$, $R$ such that $A = QR$. What is the fastest way to find $Q'$ and ...

**5**

votes

**0**answers

200 views

### Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia):
$$
\lambda^\downarrow(A) \cdot ...

**1**

vote

**1**answer

136 views

### Are there good ways of relating a minor to the full determinant?

Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...

**-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.

**2**

votes

**0**answers

77 views

### Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?

Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.)
I want to calculate or bound $\det ...

**1**

vote

**0**answers

69 views

### Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix

I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...

**5**

votes

**0**answers

278 views

### Transforming a binary matrix into triangular form using permutation matrices

I am interested in the complexity of the following problem:
Given an $m\times n$ binary matrix $M$, can we permute its rows/columns to obtain a triangular matrix?
I am also interested in ...

**1**

vote

**1**answer

120 views

### The norm of a Finite Hilbert matrix

Let $H$ be an $n\times n$ Hilbert matrix,
$$h_{ij}=(i+j-1)^{-1}.$$
The matrix $p$-norm corresponding to the p-norm for vectors is:
$\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left ...

**12**

votes

**3**answers

878 views

### How do we show this matrix has full rank?

I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988):
Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
...

**5**

votes

**1**answer

162 views

### Finding sparsest solution of a linear system

I want to find the solution with most zero-components for the following problem:
$Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...

**1**

vote

**1**answer

50 views

### Maximizing a certain concave function over a non-convex set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...