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Questions about the properties of vector spaces and linear transformations, including linear systems in general.

4
votes
1answer
46 views

On a special type of normed linear spaces

Let $(V,||.||)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $||f(x*y)||\ge ||f(x)+f(y)||,\forall x,y\in G$, is a group homomorphism i.e. $f(x*y)=...
1
vote
0answers
47 views

Linear dependence of solution?

Consider the function $f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$ In the following we want to study the ...
-3
votes
0answers
49 views

A linear algebraic question [on hold]

Let $S$ be a semigroup of $\mathbb{R}$, that is, for all $a$ and $b$ from $S$, we have $a+b\in S$, and $0$ belongs to $S$. Let $\langle S\rangle_\mathbb{Q}$ denote the vector space over $\mathbb{...
-5
votes
0answers
34 views

prove that, for every markov matrix M, there exists a vector x such that Mx = x [on hold]

(A Markov matrix is a square matrix whose columns are probability vectors) This proof I found seems simple at first but I can't understand the last step. If you have another proof, it would be ...
-1
votes
0answers
41 views

How to calculate partial derivative of a Hadamard product?

I'm trying to optimize a backpagation rule for a neural network. it involves a hadamard product but I'm not sure if this is correct. Here is my calculation: $$ x_0 = f(x) \odot g(y) $$ $$ \hat y = ...
1
vote
1answer
56 views

Lanczos algorithm for finding $k$ smallest eigenvector

I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
3
votes
3answers
179 views

Extending a continuous map over projective space

Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
3
votes
1answer
139 views

If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [duplicate]

Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
-2
votes
1answer
87 views

What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]

Suppose we have the following symmetric matrix. $$A = \sigma^2 I + u u^T$$ What can we say about the eigendecomposition of $A$?
6
votes
0answers
152 views

A linear algebra problem in positive characteristic (2)

Let $A$ be an $n\times n$ symmetric matrix with $0,1$ entries with all diagonal entries equal to $1$. Suppose $p>2$ an arbitrary prime number. Does always there exists $x \in (\mathbb{Z}/p\mathbb{Z}...
4
votes
2answers
115 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$. Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
17
votes
1answer
1k views

A linear algebra problem in positive characteristic

Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...
4
votes
0answers
78 views

Ref. request: Enumerating elements of Bruhat cells

Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then $$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$ where we embed the ...
2
votes
2answers
214 views

Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently? $$(D + ADA^T)^{-1}$$ Hints or references are ...
7
votes
1answer
121 views

Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
12
votes
3answers
462 views

Determining if some permutation of a vector satisfies a system of linear equations

Let $A$ be a matrix and $x$ a fixed vector. How can we determine whether or not there exists a permutation matrix $P$ such that $APx=0$? Does this problem reduce to anything well-understood?
5
votes
1answer
202 views

invertible endomorphisms on a space of linear maps between finite-dimensional vector spaces

This is a linear algebra question that came up in my research, and I feel like there ought to be either a simple proof or a simple counterexample, but I have been unable to find either. Assume $V$ ...
0
votes
0answers
142 views

How do we compute the trace over the matrix logarithm $\log((\sigma_2 \otimes I_{n/2})^T\cdot\Omega)$?

How do we explicitly compute the curvature form $\Omega$ of the Levi-Civita connection $\nabla^{L.C.}$ for the $n$-sphere $S^n$? Thus, how do we calculate the trace over the matrix logarithm $\log(...
1
vote
0answers
46 views

Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D

I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation: $AXB + (AXB)^T + cX = D$ where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...
0
votes
0answers
40 views

Blind signal separation for sparse signals [migrated]

Assume we have $N$ measurements $z_1, ..., z_N \in \mathbb{R}^{n_z} $ that generated by $$ z_i = M v_i + e_i $$ where $v_i \in \mathbb{R}^{n_v}$, $n_v < n_z$ and $e_i$ an error sampled from ...
2
votes
0answers
106 views

When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is a cross-post to the question I asked at MSE. Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...
0
votes
0answers
45 views

Set of functions orthogonal to $ (a - b x)^{c_n} $ [closed]

What is $v_n(x)$, s.t. $\int_{-1}^{+1} v_n(x) u_n(x) dx = \delta_{nm}$ or $\int_{-1}^{+1} v(k', x) u(k, x) dx = \delta(k-k')$, with $u_n(x) = (a-b x)^{c_n}$, $c_n$ discrete in the first, ...
0
votes
0answers
97 views

Integer reduction of a positive semidefinite matrix

Take $\Sigma $ a real positive semidefinite matrix. Define $P$ to be the smallest projection with the property that for any $\mathbf{a}\in \mathbb{Z}^n$ with $\mathbf{a}^\dagger (I-P)^\dagger \Sigma (...
0
votes
0answers
24 views

Simultaneous movement toward barycenters - what can be guaranteed

Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex. Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...
4
votes
0answers
105 views

What is the Jarlskog invariant, conceptually?

Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity: $$J_{ij,k\ell} := \operatorname{...
5
votes
3answers
355 views

Irreducible representations and invariant subspaces

Consider two invertible n-by-n matrices, $n>2$, $X$ and $Y$ over a finite field $k$ (say for simplicity $k=\mathbb Z/ \mathbb Z_2$). Is there any reasonable way to check that there is no proper ...
1
vote
0answers
97 views

Sufficient conditions for all eigenvalues simple in stochastic matrix

The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical Perron-Frobenius theorem. In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple. ...
2
votes
0answers
126 views

Intersection of two varieties in $\mathbf{F}_q^n$

Suppose we identify $\mathbf{F}_q^n$ with $\mathbf{F}_{q^n}$. Let $X_n$ be the irreducible hypersurface defined by $Nx=1$ where $N$ is the norm map. There is an analogous hypersurface $X_{n-1}$ in $\...
0
votes
0answers
95 views

Closed-form solution about a matrix factorization problem?

I am doubting some equations in paper "Multi-View Learning With Incomplete Views". The paper could be found here. The problem is related to Eq.(2) and Eq.(4b) in this paper. I summarize them here (...
3
votes
1answer
215 views

Error in Hoffman-Kunze (normal operators on finite-dimensional inner product space with a cyclic vector)

I'm teaching a second course in advanced linear algebra, following the second half of Hoffman-Kunze. I have come across what I believe to be an error, but I want confirmation (or refutation) by ...
4
votes
1answer
106 views

Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
5
votes
1answer
116 views

Numerical minimization spectral norm under diagonal similarity

This question is a follow up. Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e., $$ s(A) = \inf_{D} \lVert D^{-1} A D\...
0
votes
1answer
153 views

root of identity matrix and lexicographic order

I asked a question here order of a permutation and lexicographic order but it seems*** that a very powerful and rich generalization can be made! Let $A$ be a finite ring together with an arbitrary ...
5
votes
2answers
198 views

Minimize spectral norm under diagonal similarity

Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e., $$ s(A) = \min_{D} \lVert D^{-1} A D\rVert_2, $$ where $D$ ...
-1
votes
0answers
24 views

Finding numbers that double when you switch the first and last digit [migrated]

So, I was watching this video by MindYourDecisions about finding the smallest number that doubles when you move the last digit to become the first digit. Actually, I just saw the title of the video, ...
2
votes
1answer
68 views

Is a simple graph matrix the sum of a “shiftordered” matrix and its transposed matrix

This is the generalization of a question Is a simple graph the "sum" of a partial order and its dual? Nik Weaver found a counterexample in a very nice, complete (and instantaneous!) answer,...
4
votes
2answers
304 views

Is a simple graph the “sum” of a partial order and its dual?

A "$n$-order matrix" $T\in M_n(\mathbb F_2)$ is a matrix such that there exists a partial ordered relation $\leq_T\subset [1,n]^2$ such that : $T_{ij}=1\Leftrightarrow i\leq_T j$ (where $T_{ij}$ is ...
0
votes
0answers
84 views

norm and conorm of elliptic cocycle be different

Let $(M,\mathcal{B},\mu)$ be a probability space and $f:M \rightarrow M$ be a measure preserving map.Let $A:M \rightarrow SL(2,\mathcal{R})$be a measurable function with value invertiable $2\times2$...
2
votes
0answers
108 views

lower bounding the absolute value of a determinant

In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or ...
0
votes
1answer
81 views

How to decompose a matrix into its orthogonal and diagonal parts (assuming it has that form)? [closed]

Assume that $A = U * S$ for $U$ orthogonal and $S$ diagonal, ordered and positive. If I only know $A$, is it possible to obtain $U$ and $S$? My first guess would be taking the singular value ...
0
votes
0answers
157 views

Are the integers a vector space or algebra over “some” field or over “some” ring?

Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...
1
vote
1answer
170 views

Is there a generalization of eigenvalues and eigenvectors to tensors?

Two perhaps ill-posed or just silly questions: Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental ...
2
votes
1answer
78 views

Local-Global Principle in Graph Spectrum

The question is a bit vague, but any ideas/directions will be appreciated. Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ ...
4
votes
0answers
64 views

Monoid cohomology of $\mathbb{N}$ for a linear algebraic group

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...
1
vote
1answer
57 views

Conditions to obtain a real logarithm of a unitary unimodular complex matrix?

The problem statement is the following: $$U=\exp\{iV\}$$ where $U$ is a unitary unimodular matrix of the following form: $$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
2
votes
1answer
114 views

Controlling angles between vectors using sum of subvector angles?

This is a technical question coming out of my research. Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that $$ \...
1
vote
0answers
71 views

Decomposition of Determinant of Sub-Matrices of a Matrix

Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means $$ {\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...
1
vote
0answers
59 views

Decomposition of a Matrix by Sparse Matrices

Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$. $\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...
1
vote
0answers
25 views

find linear approximation of non-linear matrix transform [closed]

I have a square matrix denoted as $A$ and an element-wise square operator $\sigma$, such that $\sigma(A)=a_{ij}^2$,$\forall i,j$, $a_{ij}$ is the ith row and jth column element of $A$. Is there exists ...
0
votes
0answers
24 views

Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...