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

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3
votes
3answers
565 views

Algebraic K-theory can be seen as a generalization of Linear algebra? [closed]

Algebraic K-theory can be seen as a generalization of Linear algebra? If yes, how so?
8
votes
1answer
123 views

A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem. Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
2
votes
0answers
143 views

Cavalieri's principle and inversion of the Vandermonde matrix

There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...
0
votes
0answers
118 views

classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...
8
votes
2answers
474 views

Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation. Since I essentially need $n\le 4$, I think that I can show it ...
1
vote
0answers
139 views

Is finding a single vector in the null space as difficult as discovering the whole null space?

Let $A \in \mathbb R^{k\times n}$ be a matrix of rank $k$, where $k \ll n$. One can use Gaussian eliminations to discover $\operatorname{null}(A)$ at the cost of $O(nk^2)$. My question is: Is the ...
3
votes
0answers
97 views

Rank 1 Approximation of Elementwise Inverse Matrix

I'm wondering whether there is a good way to solve the following optimisation problem. Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that $$ \| D_1 A ...
0
votes
0answers
68 views

Heat equation with graph laplacian

I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix. Suppose we have a connected graph with unknown temperature on vertices. ...
1
vote
1answer
124 views

Position of complete flags

$E$ is a vector space of dimension $n \geq 2$. $\mathbb{F}=(F_1,F_2,\dots,F_n)$ and $\mathbb{G}=(G_1,G_2,\dots,G_n)$ are two complete flags of $E$. We say that $(\mathbb{F},\mathbb{G})$ is in ...
0
votes
0answers
17 views

C-Periodic boundary conditions

I'm working with linear chain of strongly correlated electrons. These types of models have problems due to finite size effects, this leads one to consider C-Periodic boundary conditions as an attempt ...
4
votes
1answer
285 views

dim Hom(V,W) =?

I asked this question on Mathematics Stack Exchange, but got no answer: Given two vector spaces $V$ and $W$ over a field $K$, what is the dimension of $\operatorname{Hom}_K(V,W)\ $? To state the ...
13
votes
1answer
383 views

A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space

While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers: Question: Do there exist a non-trivial ...
1
vote
0answers
34 views

Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]

I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by ...
7
votes
1answer
274 views

What is the total polarization of the determinant?

Let $A\in\mathfrak{gl}(\mathbb{R},n)$ be an endomorphism, and think up to conformal factors (in particular, $\Lambda^n\mathbb{R}^n$ will be the same as $\mathbb{R}$). By the total polarization ...
2
votes
3answers
263 views

Solving a quadratic matrix equation with non-squared matrix

I was trying to solve the problem of finding the value of a non-squared matrix $T$ ($n \times m$) which solves $$ T^T T = X$$ where $X$ is a symmetric and positive semidefinite $m \times m$ matrix, ...
0
votes
1answer
196 views

Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal. In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...
0
votes
0answers
86 views

Perturbed linear system, particular form

We have a linear system $Ax=b$ where $A$ is real and symmetric, all elements of its main diagonal are strictly positive and all off-diagonal elements are $\leq 0$. Further, $A_{ii} > -A_{ij} \; ...
6
votes
2answers
306 views

$\|T\|_2 \le \sqrt{\|T\|_1\|T\|_\infty}$

Let $T$ be a linear operator acting on a finite-dimensional real or complex vector space. As a direct consequence (or rather a particular case) of the Riesz-Thorin theorem, we have $$ \|T\|_2 \le ...
1
vote
0answers
69 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., ...
1
vote
0answers
76 views

Bounds for the infinity norm of the inverse for certain diagonaly dominant matrices

I m trying to analyse the stability against perturbations for a specific system of linear equations $Ax=b$. For this, i use the standard condition number $||A||_{\infty}||A^{-1}||_{\infty}$. Here ...
3
votes
1answer
135 views

eigenvalues of product of many symmetric positive definite matrices

Given $A_1, ..., A_n$ ($n\geq 3$), where each $A_i$ is a $d$-by-$d$ symmetric, positive definite matrix, define $S = A_1\cdot A_2\cdot...\cdot A_n$ (product of all the $A_i$'s). Let $\lambda_1(A)$ and ...
3
votes
1answer
361 views

Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...
1
vote
1answer
45 views

Maximising a Rayleigh quotient over a subspace II

Let $M$ be an SPD matrix and let $\Pi=QQ^T$ be the orthogonal projection onto the range of $Q$ (a "tall" matrix with orthonormal columns). I have an expression in the form $$\tag{1} ...
2
votes
0answers
100 views

Hilbert Schmidt Operators and the Conditional Expectation Operator

Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
0
votes
0answers
56 views

Trying to solve for total derivatives at a stationary point (maybe using the implicit function theorem)

Suppose we have a function $F(q) \in \mathbb{R}$, where $q=(q_1, \dots, q_n) \in [0,1]^n$, at least thrice differentiable in $(0,1)^n$. We fix the value of one variable $q_i \in (0,1)$, then maximize ...
2
votes
0answers
60 views

Characterize the equivalence class of bipartite graphs obtained from each other by elementary row operations on their adjacency matrices

Let $M$ be an $m\times n$ matrix real matrix. Let $G$ be a bipartite graph, with partitions $A$ and $B$, such that $|A|=m$ and $|B|=n$. A node $i\in A$ is linked to a node $j\in B$ if and only if ...
1
vote
1answer
66 views

Maximising a Rayleigh quotient over a subspace

Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...
2
votes
0answers
33 views

Metabolic vs stably metabolic

Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ ...
3
votes
0answers
102 views

Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows: For a ...
0
votes
1answer
87 views

Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$. The goal is to return $\widehat A, \widehat B$ such ...
1
vote
0answers
26 views

Heuristic for choosing n-vectors from n-sets

my given problem is: choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...
2
votes
1answer
75 views

Triangular Smoothing Formula Optimization

I'm using a 5-point Triangle Moving Average: $$S_j = (Y_{j-2} + 2Y_{j-1} + 3Y_j + 2Y_{j+1} + Y_{j+2}) / 9$$ The problem is that I often need to smooth my data more than once, and when I do this too ...
0
votes
0answers
41 views

Spanning Hadamard product powers (Schur products) in Euclidean space

This is a question I asked last year over at stackexchange. Got no answer, it might be sufficiently non-trivial to justify bringing it into here. Fix two $k$-vectors $\mathbf u$ and $\mathbf v$, and ...
3
votes
4answers
342 views

Reference for an algebraic group preserving a cubic form

Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...
4
votes
0answers
91 views

A question on hyperplanes in partial linear spaces and hypergraphs

A partial linear space (or a linear hypergraph) is a point line geometry $(P,L,I)$ where for every pair of points there is at most one line incident with both of them. A hyperplane in a partial linear ...
0
votes
1answer
69 views

Decompositions of sparse symmetric matrices and methods for solving large linear equations

I am writing code for solving linear equations of the form $$A_{n\times n}\cdot x=1_n$$ where $n$ is on the order of $10^6$ and $A$ is a symmetric matrix with approx $10^3$ nonzero entries in each ...
6
votes
2answers
254 views

Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$. Does an upper ...
6
votes
1answer
194 views

Solving for a set of points from projections

Consider a set of $N$ points in $n$-dimensional space, i.e. \begin{align*} S = \{x_1, \dots, x_N\} \subset \mathbb R^n. \end{align*} Let $v \in \mathbb R^n$ and consider the image set (not counting ...
1
vote
1answer
77 views

Bound on the 2-norm of a “special” matrix

Let $S\in\mathbb{R}^{n\times n}$ be such that $\|S\|_2\leq 1$, $P\in\mathbb{R}^{n\times m}$, $m<n$, with orthogonal columns ($P^TP=I$) so that $PP^T$ and $I-PP^T$ are orthogonal projectors, and ...
1
vote
2answers
254 views

when will the surfficient large power of a rational matrix be a integer matrix?

$A$ is a $n\times n$ matrix whose elements are all non-negative rational numbers and $Det(A)$ is a non-zero integer.Under what condition the following is true?(0) There exist a positive integer $M$ ...
2
votes
1answer
84 views

General method for under and over determined systems?

Suppose I have a system: $$ Ax = b $$ where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$. I'd like a combination of a ...
2
votes
0answers
303 views

The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE) While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
0
votes
0answers
48 views

Linear Bounds on estimation error

Consider a markov chain on discrete state space $\mathbb{S} = \left\{1,2,..,S \right\}$, with transition probability matrix defined as $A = [a_{ij}]_{S \times S}$ where $a_{ij} = ...
-2
votes
1answer
90 views

Is dual cone unique? [closed]

Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones, $A^\ast=C,$ $B^\ast=C,$ can we state that $A=B$? Is the dual cone of a cone is unique? the definition ...
2
votes
2answers
343 views

Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v1 §19)

$\newcommand{\refone}{\textbf{(1)}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\tr}{\operatorname{Tr}}$ $\newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which ...
1
vote
0answers
64 views

Dominant eigenvalue of sum of tridiagonal and diagonal matrices

Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except ...
2
votes
1answer
72 views

An algebraic equation question [closed]

My question is this: If $\frac{\sqrt[n]{\prod_{i=1}^n(p_i + 1)}}{\sqrt[n]{\prod_{i=1}^n(m_i + 1)}} = e ^\beta$ can I find an expression (either exact or approximate) for ...
2
votes
0answers
52 views

Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings

In one paper from 1980 I found a note that there are no known algorithms for solving homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring. (The ...
0
votes
1answer
69 views

Eigendecomposition of a summation of matrices [duplicate]

Can anyone tell me if there's a way to relate the eigendecomposition of the result of a summation of matrices with the eigendecomposition of those matrices? More specifically: If I have a matrix $K = ...
7
votes
1answer
239 views

Casson invariant

Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does ...