**0**

votes

**0**answers

8 views

### force singular value decomposition to use the solution with the biggest absolute value

Well I'm writing a code to solve a positioning problem. given arrival times from multiple sources I want to invert and get the receiver position. obviously I have the xyz of each receiver.
so I ...

**4**

votes

**2**answers

77 views

### Why is the spectrum of this matrix product invariant with respect to order of the multiplicants?

I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A_1,...,A_k \in \mathbb R^{n\times n}$ and I'm observing that the ...

**9**

votes

**1**answer

116 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

**0**answers

139 views

### Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$
Let me give a ...

**7**

votes

**1**answer

96 views

### Compute only selected components of an eigenvector

I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...

**4**

votes

**1**answer

153 views

### $n$ columns of a specific “infinite” Vandermonde matrix always linearly independent? [on hold]

Consider the "infinite" Vandermonde matrix
$$
V (x_1, x_2, \ldots , x_n) =
\begin{pmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\
...

**0**

votes

**0**answers

28 views

### basis span of space necessary to be orthogonal? [on hold]

If a vector space V that span of {v1,v2,....,vk},can the basis vector of V are not mutually orthogonal?
For vector space V, are there exist set of orthogonal basis {o1,o2,....,ok} that can span V? ...

**0**

votes

**0**answers

53 views

### Elementary Linear Algebra Maximization Problem [on hold]

Can someone show me the proof for the following:
min $\frac{q^TLq}{q^TWq}$ where q is not 0 and subject to $qWe =0$ is solved when
q is the eigenvector corresponding to the 2nd smallest eigenvalue ...

**-1**

votes

**0**answers

60 views

### Why solving a system of linear equation produces the intersection of the equation [closed]

Let us consider two equations
1)x+y=1
2)-x+y=1
Consider the solution of the equations using Gaussian Elimination,
Geometrically I am able to understand the ...

**3**

votes

**1**answer

93 views

### Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$

I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...

**0**

votes

**0**answers

34 views

### Derivative of softmax loss function [closed]

I am trying to wrap my head around backpropagation in a neural network with a softmax classifier, which uses the softmax function:
\begin{equation}
p_j = \frac{e^o_j}{\sum_k e^{o_k}}
\end{equation}
...

**-1**

votes

**1**answer

73 views

### Action of rotation group on Matrices [closed]

Is the following assertion true?
Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in ...

**-3**

votes

**1**answer

117 views

### A problem that involves matrix and Lorentz Transformation [closed]

To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes.
$1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G ...

**-2**

votes

**0**answers

60 views

### How to plot 2-D and 3-D Joint numerical range of real symmetric matrices in mathematica? [closed]

I know this question this doesn't belong here. However, I am not getting a satisfactory reply from the mathematica forum. Consider $2\times 2$ real symmetric matrices $\mathbf{A}_1$ and ...

**1**

vote

**0**answers

40 views

### Convergence to Eigenvalue gap of Gaussian Orthogonal Ensemble

Let $M_n$ come from the Gaussian Orthogonal Ensemble of size $n\times n$. Let $E_1(0; I)$ be the probability that $M_n$ has no eigenvalues in an interval $I$. The bulk scaling limit of this is defined ...

**1**

vote

**1**answer

67 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

110 views

### Is $Rank(A)=Rank(A^{T})$ where $A$ has inifinite rows and columns and its given that $A$ has finite rank [closed]

Is $Rank(A)=Rank(A^{T})$ where $A$ has inifinite rows and columns and its given that $A$ has finite rank.
Is it a sub-case of finite rank operators which maps finite dimensional compact sets(which ...

**1**

vote

**2**answers

100 views

### The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...

**3**

votes

**0**answers

52 views

### Dimension of the sum of images of transpose

$\newcommand{\rank}{\operatorname{rank}}\newcommand{\im}{\operatorname{im}}$
Given $A,B\in M_{n\times n}(k)$, define $\rank(A,B):=\dim(\im A+\im B)$. I'm looking for results regarding relationships ...

**9**

votes

**1**answer

211 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 ...

**2**

votes

**1**answer

56 views

### Solution to generalized Sylvester equation

I am interested in solving generalized Sylvester equations (for $X$) of the form:
$$ \sum_{j=1}^k A_j X B_j^T = F, $$
where $A_j,B_j,X,F\in\mathbb{C}^{n\times n}$ and $k$, $n$ are integers. I will ...

**1**

vote

**0**answers

76 views

### Reference request on operator matrices [closed]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that
$$Tx = \begin{pmatrix}A & B \\
C & D
...

**0**

votes

**1**answer

89 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 ...

**0**

votes

**0**answers

56 views

### Speed up Linear programming

I have a linear programming problem like this:
minimize $c^t X$
under the constraint that $AX \ge b$.
I will need to solve this linear programming problem online many times. I need it to be as fast ...

**0**

votes

**0**answers

25 views

### Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$.
I am able to write each event as a sum of distinct events that form a partition of the space.
My goal is to find all the ...

**1**

vote

**1**answer

94 views

### Checking the intersection of two sets

Let $E\subset{\mathbb R}^n$ be a set of the type $I_1\times \dots \times I_n$, where $I_k$ are real intervals, and $X$ be and $n\times p$ real matrix. Suppose also that $rank(X)=p$ and $n>p$. Is ...

**0**

votes

**0**answers

34 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 ...

**0**

votes

**0**answers

45 views

### Applying a linear operator to a basis set following SVD orthonormalization

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases.
My intention is to:
decompose $\Phi$ via SVD:
$U \Lambda V^T = \Phi$
to create it's ...

**-2**

votes

**0**answers

26 views

### non-negativity of cost function

For which set of real parameters $\alpha $ and $\beta$ the following
expression is negative and for which positive
$\frac{1}{\alpha \beta} \sum_{i=1}^n \log \frac{\alpha \lambda_i^{\beta}+\beta ...

**1**

vote

**0**answers

64 views

### A criteria for a subalgebra of M(n,C) being M(n,C) [migrated]

Suppose $S$ is a subalgebra of the matrix algebra $M_n(\mathbb{C})$. If for any vector $v$ and $w$ in $\mathbb{C}$, there always exists a matrix $A$ in $S$, depending on $v$ and $w$ of course, which ...

**1**

vote

**0**answers

73 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,
...

**5**

votes

**0**answers

187 views

### Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...

**9**

votes

**0**answers

189 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 ...

**0**

votes

**1**answer

83 views

### Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space

Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.
I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...

**0**

votes

**0**answers

67 views

### determinant inequality for symmetric positive definite matrices

Assume that A and B are symmetric, positive definite matrices of the same size.
For which set of real parameters $\alpha $ and $\beta$ the following relation holds
...

**3**

votes

**1**answer

111 views

### Two matrix Fisher distributions on SO(3)?

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...

**19**

votes

**1**answer

792 views

### Recognize this strange expression from linear algebra?

I've come across an odd-looking expression and oh how I wish I had a more elegant description of it. Maybe someone who enjoys symmetric bilinear forms in characteristic two will recognize it? Or ...

**0**

votes

**2**answers

111 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

166 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?

**0**

votes

**1**answer

57 views

### Schur norm for self-adjoint operators

If $A$ is a $n \times n$ complex matrix then the Schur norm of $A$ is given by $$ || A||_S := \max_{||B||=1} ||A*B||,$$ where $||. ||$ is the operator norm and $*$ is the Hadamard (entry-wise) ...

**4**

votes

**1**answer

45 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, ...

**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 ...

**0**

votes

**1**answer

110 views

### Dimension of Commutator Space [closed]

For each $n\times n$ matrix $A$ with real entries the set
$$C(A)=\{X\in M_n(\mathbb{R}): AX=XA\}$$
is obviously a linear subspace of $M_n(\mathbb{R})$.
Can we recognize the dimension of this ...

**0**

votes

**1**answer

125 views

### Proving that the eigenvalues of a certain matrix product are positive

Let $A$ be an $m \times n$ matrix, and define:
\begin{align*}
U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\
V &= {\rm diag} \{ \frac{1}{\alpha_i} ...

**2**

votes

**0**answers

54 views

### Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent
\begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation}
of a 3-tensor $W_{j,k,l}$ of ...

**0**

votes

**1**answer

38 views

### Adjusting matrix in generalized eigenvalue problem for the design of eigenfunctions

Take two matrices $A,B \in \mathbb{R}^{n\times n}$. I am considering the generalized eigenvalue problem ($\gamma \in \mathbb{R}, \phi \in \mathbb{R}^n$)
$$A\phi = \gamma B\phi.$$
Is there a ...

**7**

votes

**1**answer

191 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 & ...

**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 ...

**3**

votes

**1**answer

98 views

### $\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.
Moreover, let $M \colon \mathbb{R}^n \to ...

**2**

votes

**0**answers

70 views

### Separating duality for TVS?

What is the modern concept (term) for "separating duality" (dualité séparante in french) in the sense of Bourbaki (TVS Ch II § 6) as explained in the following ?
...