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When does a real-valued function of a matrix depend only on eigenvalues?

Let $\mathcal{N}$ be the space of all $n \times n$ matrices that are similar to some nonnegative matrix with zero diagonal and let $f: \mathcal{N} \to \mathbb{R}$ be a continuously differentiable ...
Ben Golub's user avatar
  • 1,068
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0 answers
244 views

Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of ...
io0's user avatar
  • 1
0 votes
1 answer
2k views

Finding linearly independent columns of a large sparse rectangular matrix

I have a problem that necessitates solving a large non-negative least-squares problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols) and nearly binary. However, A is not ...
Rob's user avatar
  • 103
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0 answers
325 views

Changing basis on an extension of a free Z-module.

Consider a finite-rank free $Z$-module $Y$. Let $c: Y \times Y \rightarrow Z$ be a $Z$-bilinear form. Assume that $c(y_1, y_2) + c(y_2, y_1)$ is even, for all $y_1, y_2 \in $. Then $c$ "incarnates"...
Marty's user avatar
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0 answers
154 views

linsolve derivative

Consider a vector $\mathbf{g} \in \mathbb{R}^{m}$ and a matrix $\mathbf{A} \equiv \mathbf{A(g)} \in \mathcal{M}_{p\times q} \[\mathbb{R}\]$, a function of $\mathbf{g}$. Furthermore, let $\mathbf{S} \...
user25779's user avatar
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0 answers
204 views

Matrix Mutiplication through Matrix Logarithms and Exponentials

Let $A,B$ be full rank $n \times n$ matrices. If $AB = BA$, then $\exp(\log(A)+\log(B))=AB$. Supposing $A = USL$ and $B = VSL$ where $U,V,S,L$ are integer valued matrices, $det(L)=1$ and $U = LVL^{-1}...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
161 views

vector equation

Suppose you have an equation of the form $Hx=Ky$, where $x,y$ are vectors of length $n,m$ respectively ($m>n$) and $H,K$ are matrices of orders $n \times n,n \times m$ respectively. Is there some ...
Felix Goldberg's user avatar
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0 answers
138 views

Approximation of large dimensional vectors by vectors of smaller dimension

sIs there any (efficient) algorithm for the following problem: Let $n = 128$ and $m = 64$ (in the end only $n > m$ matters) and $p_1, \ldots, p_t \in \{ -1,1 \} ^{128}$ be given ($t << 2^{...
tobias's user avatar
  • 397
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0 answers
276 views

Another matrix diagonalization problem

Given the matrices $X$ and $Y$ in $[0,1]^{n\times m}$, for $n > m > 3$, so that $X1_m=1_m$ and $Y1_m=1_m$, where $1_m$ denotes a $m$-length column vector of ones, find a matrix $Q$ in $R^{m\...
silvanmx's user avatar
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0 answers
395 views

The ratio of two strictly increasing functions

Given: \begin{equation} f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i \end{equation} \begin{equation} f_2(a)=\sum_{i=1}^{k^*-1} ...
Seyhmus Güngören's user avatar
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0 answers
109 views

Expansion (asymptotic) of scalar function of a square matrix , in terms of determinant of argument?

The title says it all. I have a scalar function (really, a determinant) of a square matrix argument. Can I find an (asymptotic) expansion of the function, in a series in the determinant of the ...
kjetil b halvorsen's user avatar
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0 answers
2k views

In a network with N nodes, what is the general formula for computing the propagation of a set of numbers?

I am creating a circular neural network with N nodes. Each node is connected via a send pathway to every other node, and the connection between two nodes has a weight. Any number sent over the ...
AIGuy's user avatar
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0 votes
1 answer
655 views

Fuzzy vector similarity

Hi all, I have two multi-dimensional vectors representing documents $\vec{a}$ and $\vec{b}$. Considering cases where there is no overlap between $a$ and $b$ ($a \cap b = \emptyset $), traditional ...
user17528's user avatar
  • 103
0 votes
1 answer
130 views

Maximal length vector under constraints

Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
user16007's user avatar
  • 800
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0 answers
430 views

[]-infinity algebra and Projective representation

This is a very vague question. We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...
Ma Ming's user avatar
  • 1,271
0 votes
0 answers
158 views

Matrices satisfying certain pair-wise constraints

Consider given pairs of variables: $a_{ir1},a_{ir2}\in \mathbb{R}^{m \times m}$ and $a_{jr1},a_{jr2}\in \mathbb{R}^{m \times m}$, where $r \in \{1,2,\cdots,t\}$, consider the constraints: $\sum_{r=1}^...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
172 views

Generating Set for $O(V)$ over $\mathbb Z_2$

I am reading a claim that $O(V)$ — the orthogonal group associated with a finite-dimensional vector space $V$ over $\mathbb Z_2$ and a quadratic form $q$, i.e. the group of linear ...
Larry's user avatar
  • 105
0 votes
1 answer
503 views

When are operators extended by linearity bounded?

Greetings. Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
Adam Azzam's user avatar
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0 answers
352 views

Liftability in positive characteristic

What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
Universe's user avatar
0 votes
0 answers
524 views

DeRham cohomology

The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
chemaida's user avatar
0 votes
0 answers
1k views

Determinant of special generalized Vandermonde matrix

Good evening! I have a generalized Vandermonde matrix of special form: $\left( \begin{array}{ccccc} a_{0,0} & a_{0,1} \cdot x_0 & a_{0,2} \cdot x_0^2 & \ldots & a_{0,m-1} \cdot x_0^{m-...
user avatar
0 votes
0 answers
608 views

Orthogonal Projections in Lie Theory

I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
John Craighead's user avatar
0 votes
1 answer
153 views

Difference of two optimization problem's optimal value

Let we have two following optimization problems: \begin{align} \text{(P1)}\quad \alpha_1 = \max_{x_1,\ldots,x_M} &\quad \sum_{m=1}^{M}\log(1+f_m(x_1,\ldots,x_M))\\ \textrm{s.t.} &\quad \...
Math_Y's user avatar
  • 287
0 votes
1 answer
199 views

Intersection between a line and an n-dimensional parallelotope

Suppose that I have a line in an $n$-dimensional space described by $$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$ here $A$ is known and I want to find all the possible vectors $B$ ...
Leonardo's user avatar
0 votes
1 answer
268 views

Nonnegative Matrix

Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ${\...
Yiyan's user avatar
  • 303
0 votes
1 answer
180 views

(probably simple) optimization question

Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
Felix Goldberg's user avatar
0 votes
2 answers
2k views

How to accelerate/avoid multiplication for large matrices in Matlab? [closed]

The setting is here. X: 6000x8000 non-sparse matrix B: 8000x1 sparse vector with only tens of non-zeros d: positive number M: is sparsified X'X, i.e. thresholding the elements smaller than d ...
Peter's user avatar
  • 21
0 votes
2 answers
372 views

Quantum observables

Let H be a Hilbert space and A, B two non-commuting bounded linear operators. Let Com(A,B) be the set of bounded linear operators C which commute both with A and B. Question 1 : What is known about ...
Elemer E Rosinger's user avatar
-1 votes
1 answer
681 views

Is there such a thing [closed]

For any $U_{i}\in\mathcal{U}\left(4\right)$, $1\le i\le5$, are there $W\in\mathcal{U}\left(4\right)$ and nontrivial $\left(x_{1},x_{2}\right)\in\mathbb{R}^{2}$, such that $\mbox{tr}\left(U_{i}\mbox{...
sophie's user avatar
  • 53
-1 votes
1 answer
305 views

A simple matrix multiplication query [closed]

The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given ...
Turbo's user avatar
  • 13.9k
-1 votes
2 answers
336 views

How to generate constant row and column sum matrices?

How can we randomly generate matrix $A \in \mathbb{R}^{n \times m}_{\geq 0}$ that satisfies $A 1_n = m1$ and $A^T 1_m = n1$.
seg nana's user avatar
-1 votes
1 answer
809 views

On an eigenvalue inequality [closed]

Let $\lambda_1 (\cdot)$ be the larger absolute value eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$ the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e. $|\lambda_1 (\cdot)| \...
user20216's user avatar
-1 votes
2 answers
806 views

The lie algebra of the orthogonal group of an arbitrary space time metric

Let X ad Y be two vectors in R4, and define the inner product of X and Y as: (X*Y) = gikXiYk (summation convention for repeated indicies) Then we consider the 4x4 matrix g whose components are gik. ...
Matt's user avatar
  • 251
-1 votes
1 answer
152 views

Topological characterization of invertible real matrices [closed]

Let $n\geq 2$ be an integer. Consider the topological space $M_n$ of $n$-by-$n$ matrices with real entries. Can you give a short non-constructive proof of the existence of a continuous function $M_n\...
orname's user avatar
  • 23
-1 votes
1 answer
185 views

eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $

Let $A$, $B$ and $C$ be symmetric matrices. What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
Moh514's user avatar
  • 461
-1 votes
1 answer
248 views

Maximal commutative subrings of the endomorphism ring of a vector space

Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
Bedovlat's user avatar
  • 1,959
-1 votes
1 answer
172 views

$A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?

My question follows from https://math.stackexchange.com/questions/3857976/inverse-inequality-of-symmetric-matrix. Suppose we assume that $A$ and $B$ are two positive definite matrices with positive ...
user812951's user avatar
-1 votes
1 answer
215 views

Dense linear span implies closed convex hull has non-empty interior

Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
ABIM's user avatar
  • 5,405
-1 votes
1 answer
195 views

Determinant of $Z^TZ$ [closed]

If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
gradstudent's user avatar
  • 2,246
-1 votes
1 answer
121 views

On rank one torsion-free modules over local rings [closed]

Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes_A k \le 1$? If not, is there a known upper-bound ...
Ron's user avatar
  • 2,126
-1 votes
1 answer
1k 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.
Hao S's user avatar
  • 111
-1 votes
1 answer
142 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 SO(p,\...
Vanya's user avatar
  • 601
-1 votes
1 answer
1k views

Sum of two unitary matrix is equal to every matrix? [closed]

Let $R=M_{n}(Z_{2})$, can we write every matrices of $R$ as sum of two matrices of $GL_{n}(Z_{2})$?
Moh514's user avatar
  • 461
-1 votes
1 answer
137 views

Does a half plane contain intersection of some other half planes? [closed]

I'm doing research in Optimization and I have found this obstacle in the way. If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
Nothing's user avatar
  • 19
-1 votes
2 answers
439 views

Are the coefficients of a linear combination of random vectors as random?

Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...
Rob's user avatar
  • 271
-1 votes
1 answer
360 views

Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
Tobi's user avatar
  • 7
-1 votes
1 answer
156 views

Companion matrices must have geometric multiplicity one, linear recurrence sequence view [closed]

I posted this question on math stackexchange weeks ago, and it have not receive an answer yet after a bounty offer... I've been recently playing around with the linear recurrence sequences. Consider ...
Lab's user avatar
  • 109
-1 votes
1 answer
324 views

Expressing the sum of two squared inner products more compactly: is it possible to lift the dimension? [closed]

Let $v_1,v_2\in\mathbb{R}^d$ be two fixed vectors, and $\langle \cdot,\cdot\rangle_{\mathbb{R}^d}$ be the usual Euclidean inner product in $\mathbb{R}^d$. My question is as follows. Is there an (...
hookah's user avatar
  • 1,096
-1 votes
1 answer
62 views

Basic operation geometrical meaning [closed]

What is the geometrical meaning of doing $x^TAx \;$? $Ax \; $ is trivially "applying A to x", but then, what the multiplication for $x^T$ stands for?
Mattia Podio's user avatar
-1 votes
1 answer
132 views

About a property in a reflexive Banach space

Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) ...
MSMalekan's user avatar
  • 2,118