**0**

votes

**0**answers

41 views

### Diagonally change the matrix [on hold]

if we have a matrix
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
then we have to change the elements diagonally from top left to bottom right ? what it ...

**9**

votes

**1**answer

193 views

### What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...

**0**

votes

**1**answer

52 views

### Eigenvalues of product of diagonal positive matrix and symmetric matrix [closed]

Assume that we have two real symmetric matrices A and B, where A is a positive diagonal matrix, and B is a symmetric matrix with one eigenvalue λ = 0. Assume that H= AB;
is it possible to proof that ...

**-2**

votes

**0**answers

36 views

### How to show a matrix can't be written as exponential? [migrated]

How can I show the matrix
$$A = \left(
\begin{array}{c c}
-2 & 0 \\
0 & -1 \\
\end{array}
\right)$$
can't be written as $A = exp(a)$?
I've tried to write A like
$$A = \left(
...

**1**

vote

**1**answer

148 views

### Name for class of matrix determinants

Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $\alpha_i(x)$. Now, ...

**4**

votes

**1**answer

182 views

### Another kind of the positivity of matrices

For given $n$, the following $n\times n$ complex matrix $M=M^{\dagger}$ is called positive, if
$x^{\dagger}M x\geq 0$
holds for all complex vector $x=(z,z^2,\cdots,z^n)^T$ with arbitrary complex ...

**0**

votes

**1**answer

83 views

### Symmetric Zero-Diagonal Matrices

Consider matrices with entries in a field $F$ of characteristic $2$. Let $\Omega$ denote the $2n\times2n$ matrix $\left[\begin{array}{ll}0&1_n\\1_n&0\end{array}\right]$. Then $X^t\Omega X$ is ...

**9**

votes

**2**answers

364 views

### Are bounds known for the maximum determinant of a (0,1)-matrix of specified size and with a specifed number of 1s?

The problems of determining the maximum determinant of an $n \times n$ $(0,1)$-matrix and the spectral problem of determining exactly which other determinants can possibly occur are both reasonably ...

**0**

votes

**0**answers

32 views

### Multiplicative Symmetrization of Bilinear Forms

Let $F$ be a field of characteristic $2$, let $V$ be an even dimensional $F$-vector space, let $B$ be a non-degenerate symplectic bilinear form on $V$ and let $^*$ be the adjoint of $B$ on ${\rm ...

**-1**

votes

**0**answers

73 views

### Cayley-Hamilton type decomposition of SL(3,R) matrices [closed]

Given an element $$\lambda = \sum_{a=1}^{8}\theta_a T_a$$ of SL(3,R) Lie algebra, where $T_a$'s are the generators and $\theta_a$'s are the parameters, is there a general formula or way to decompose ...

**1**

vote

**0**answers

69 views

### Bounds on the effect of a matrix product on the Frobenius norm

I was wondering if there was a way to put upper and lower bounds on the Frobenius norm of a matrix product in relation to the Frobenios norm of one of the individual matrices, i.e,
...

**9**

votes

**0**answers

179 views

### Determinant inequality involving Hermitian, positive definite matrices

Question:
Let $A,B,C\in M_{n}(C)$ be Hermitian and positive definite matrices such that:$$A+B+C=I_{n}$$
Show that:
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question ...

**1**

vote

**0**answers

70 views

### $\epsilon$-covering number of a set of rank-2 matrices

Suppose that two unit-norm vectors $\boldsymbol{a}\in \mathbb{R}^m$ and $\boldsymbol{b}\in\mathbb{R}^n$ are given with $m\leq n$. Furthermore, let $\boldsymbol{F}_{m,n}$ denote the first $m$ rows of ...

**0**

votes

**2**answers

104 views

### Proof for a Rank-One Decomposition Theorem of Positive (semi) Definite Matrices

Consider the following result which I recently came across in a research paper in my area (Signal Processing)
Let $X$ be a $N\times N$ positive semidefinite (psd) matrix whose rank
is $r$. Let ...

**1**

vote

**1**answer

161 views

### Largest eigenvalue of the sum of hermitian matricies

Is there an expression for the largest eigenvalue of the sum of two hermitian matricies in terms of the spectrum of the same matricies?

**4**

votes

**0**answers

76 views

### Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...

**6**

votes

**2**answers

298 views

### Do compact groups acting irreducibly have finite subgroups which do the same?

Let $G$ be a closed subgroup of $U(n,{\bf C})$, not necessarily connected. Regard ${\bf C}^n$ as a complex $G$-module $M$.
Q. Suppose $M$ is irreducible as a $G$-module (equivalent, I think, to ...

**2**

votes

**0**answers

27 views

### When is the solution to a n initial value problem matrix differential equation invertible? [migrated]

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$:
$$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$
$$ ...

**1**

vote

**0**answers

44 views

### “embedding” various matrix equivalences into the equivalence of particular linear map

Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...

**2**

votes

**1**answer

93 views

### How can I prove that the negative biased triangular kernel is positive semidefinite

How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$
$k(x, x') = (1 - 2|x-x'|)$
is a positive semidefinite function?
It turns out to be psd function when ...

**0**

votes

**0**answers

63 views

### Bounding multiplications of PSD random matrices

Consider the following setup,
$(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$.
The ...

**3**

votes

**1**answer

92 views

### Smith Normal Form for block matrix

are there any known results on the smith normal form for block matrices over the integers?
In particular I am interested in matrices of size (kr)x(ks) made of square blocks of size k such that each ...

**2**

votes

**1**answer

71 views

### The convergence of Matrix factorization

I'm trying to prove the convergence of Matrix factorization.
The problem is described below.
$|X-WH|^2 + |H|_2^2 +|W|_2^2$.
My optimization steps are using Alternating least squares which update H ...

**2**

votes

**2**answers

211 views

### Gaussian expectation of an exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation,
$$ E\left[ \exp(\mathbf{xx}^\top)\right]$$
where $\exp(\cdot)$ is element-wise exponential function (not ...

**0**

votes

**1**answer

75 views

### Dimension of a similarity class

Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...

**2**

votes

**2**answers

123 views

### Asymptotic property of a quadratic form

suppose $x=\Delta$, $y=M \Phi \Delta$, where $\Delta\in N\times 1$, $M^T=M \in N \times N$ and $\Phi^T=\Phi \in N \times N$. Define $Z=xy^T+yx^T$. It is known from my previous question that $Z$ has ...

**1**

vote

**0**answers

255 views

### On increasing the penalty term in convex optimization with regularization

Given the two strictly convex (unique solution) optimization problems as:
$$Problem\:1:\min_{X} f(X)+\|X\|_{F}^2 \hspace{2cm}Problem \:2:\min_{X}f(X)+n\|X\|_2^2$$
where $X\in\mathbf{S}_{++}^{n}$ ...

**2**

votes

**1**answer

359 views

### An inequality involving traces and matrix inversions

The following question kept me wondering for some time:
Given the symmetric matrices $A,B,C\in\mathbb{R}^{n×n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...

**3**

votes

**3**answers

100 views

### Norm of the upper triangular part of symmetric matrix

Let $D\in \mathbb{R}^{n\times n}$ denote a lower triangular matrix. With $\|\cdot\|$ denoting the spectral matrix norm, is there an estimate like
$$
\|D\| \leq C\|D+D^T\|,
$$
where $C>0$ is ...

**3**

votes

**1**answer

120 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 ...

**3**

votes

**1**answer

71 views

### On the solution of a generalized Lyapunov equation

We shall reconsider the following equation
$$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$
where $p$ is a positive integer and $C$ is a known symmetric positive semidefinite matrix.
I met with this ...

**3**

votes

**1**answer

147 views

### largest eigenvalue of a symmetric matrix

I have a matrix of the form:
$X=\Delta \Delta^T (\Phi+\Phi^T) P + P (\Phi+\Phi^T) \Delta \Delta^T $,
where $\Delta$ is $N\times 1$ real, $P=P^T$. I know that such matrix is rank two, but after doing ...

**4**

votes

**1**answer

423 views

### Die hard nilpotent spaces

Let $V\subset\mathbb{C}^{n\times n}$ be a linear space consisting of $n\times n$ complex matrices. Say that $V$ is nilpotent if every matrix $v\in V$ is nilpotent; denote by $V^k$ the subspace spanned ...

**10**

votes

**1**answer

201 views

### Factor a sum of products of cofactors

Let $M$ be any $n\times n$ matrix.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write ...

**5**

votes

**1**answer

105 views

### Origins of the Jacobi matrix

I have several questions concerning history of Jacobi matrices.
Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...

**2**

votes

**0**answers

44 views

### Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...

**8**

votes

**1**answer

265 views

### Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...

**4**

votes

**2**answers

91 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

**1**answer

109 views

### How do eigenvalues change if we duplicate a row and column of a symmetric matrix

Let ${\bf A}$ be a size $n \times n$ symmetric positive semidefinite matrix with the first column being ${\bf a}_1$. If we define a new matrix,
\begin{align}
{\bf B} = \left[\begin{array}{cc} a_{11} ...

**0**

votes

**0**answers

26 views

### Proximal operator of modified L1 matrix norm

In literature proximal operator $prox_{\lambda f} : R^n \rightarrow R^n$ of $f$ is defined as:
$prox_{\lambda f}(V) = argmin(X) (f(X) + (1/2 \lambda)||X-V||^2_2)$
Consider now $g(X) = ...

**1**

vote

**1**answer

93 views

### Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...

**2**

votes

**1**answer

236 views

### Number of Matrices with bounded determinant

Here's my question:
Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...

**0**

votes

**1**answer

65 views

### Uniqueness of solution of a nonconvex optimization problem

What conditions need to be hold for a nonconvex optimization problem to have a unique solution?
Specifically, I have the following minimization problem that I'd like to know whether it has a unique ...

**4**

votes

**3**answers

215 views

### Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex
\begin{align}
\mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\}
\end{align}
and consider the ...

**0**

votes

**0**answers

36 views

### Invexity of the $L_2$ norm

I have the following function:
$ f({\bf A,b}) = \| {\bf y - XAb} \|_2^2$
where ${\bf y}_{n \times 1}$ and ${\bf X}_{n \times p}$ are fixed, and ${\bf A}_{p \times r}$ and ${\bf b}_{r,1}$ are the ...

**0**

votes

**0**answers

48 views

### Solution of a nonlinear system of two equations

Given the matrix $A_{M,N}$ with $N\gt M$, the vector $y$, I have to find the vectors $x$ and $u$, satisfying the following equations:
$$D(x)x=A^Tu$$
$$y=Ax$$
where: $$D(x) = \left| \begin{array}{ccc}
...

**0**

votes

**2**answers

91 views

### Convexity of the Frobenius norm of the product of two matrices

I have the following function for two matrices ${\bf A}$ and ${\bf B}$:
$f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$
where matrices ${\bf X}_{n \times ...

**1**

vote

**1**answer

154 views

### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...

**2**

votes

**1**answer

91 views

### Comparison of the smallest eigenvalues of two tridiagonal matrices

Let $n\geq2$ be an integer and $E_{ii}$ for an integer $2\leq i\leq n$ be the $n\times n$-matrix with its $ii$-entry equal to 1 and remaining entries equal zero. Furthermore, let ...

**0**

votes

**1**answer

74 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 ...