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

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9
votes
2answers
242 views

Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$

I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...
1
vote
1answer
60 views

Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces

Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by $$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$ $$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$ I wish to find their convex hull, that is a ...
4
votes
0answers
53 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that ...
0
votes
0answers
43 views

Find optimal value for a regularization parameter in generalized eigenvalue problem

Consider the generalized eigenvalue problem : $ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $ where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices ...
1
vote
1answer
64 views

Is there a generalization for the discrete fourier transform whereby eigenvalues are other roots of unity?

The eigenvalues of the discrete fourier transform are $\{1, -1, i, -i\}$ in approximately equal proportions. https://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors Is ...
3
votes
0answers
68 views

Spectra of certain totally positive matrices

Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions: All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix); all principal minors are $>1$, ...
12
votes
4answers
507 views

List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...
-1
votes
1answer
53 views

Computing the inverse of a Cholesky decomposition [closed]

I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ...
5
votes
2answers
261 views

Splitting subspaces and finite fields

Hellow. I'm sure that the following is truth, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and $A = ...
9
votes
2answers
224 views

Sets of matrices which are irreducible but not strongly irreducible

A set of $d \times d$ real or complex matrices is commonly called irreducible if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and $d$. A stronger ...
2
votes
1answer
184 views

Number of Plücker relations for a Grassmannian

Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: ...
2
votes
1answer
101 views

Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times ...
1
vote
0answers
96 views

Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
1
vote
1answer
50 views

Pragmatic Test for Total Unimodularity

I want perform a simple check for total unimodularity. Question: what, if anything, can be concluded from the fact, that $$det(A)=1,\ a_{ij}\in\{-1,0,+1\}\ \wedge\ a_{ij}^{-1}\in\{-1,0,+1\}$$ ...
2
votes
0answers
137 views

Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation $$ A_1 +...+A_n ...
2
votes
2answers
138 views

A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...
3
votes
0answers
83 views

Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations: $[U_i, U_j] = 0$ for $|i-j|>1$ ...
2
votes
0answers
72 views

Weyl-type inequality for non-Hermitian matrices?

What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
0
votes
1answer
48 views

Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form $A = P^TLDL^TP$, where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
2
votes
1answer
66 views

Eigenvectors of a perturbed reducible stochastic matrix

Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix ...
4
votes
2answers
151 views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
6
votes
0answers
178 views

Complexity of approximating the size of the range of a matrix

Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define: $$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$ It is NP-hard to compute $S_M$ exactly I believe by applying the ...
1
vote
1answer
148 views

Is there a nice choice-free argument to count the number of sublattices?

It's a well known fact that the number of index $n$ sublattices of a rank two lattice $\Lambda$ is given by $\sigma_1(n) = \sum_{d\mid n} d$. Here is a proof of this fact: Proof: choosing a basis of ...
10
votes
2answers
407 views

How to prove this determinant is positive-II?

Question: Given an arbitrary number of real matrices of the form $ A_i= \biggl(\begin{matrix} C_i+E_i & B_i \\ B_i^T & D_i-F_i \end{matrix} \biggr) $, where $B_i$ is an arbitrary $n\times n$ ...
1
vote
0answers
49 views

Directed graph Laplacian with exactly one negative eigenvalue

Let $G$ be a digraph with adjacency matrix $A =(A_{ij})$ where $A_{ij}=1$ if and only if there is a directed edge $i \to j$ and $A_{ij}=0$ otherwise. Let $D= (D_{ij})$ be the degree matrix with ...
-1
votes
1answer
117 views

How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? [closed]

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...
2
votes
1answer
162 views

Extracting a full rank matrix from a 0-1 matrix

If $A$ is a $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$. If ever possible, what would be an efficient way of extracting a full rank $k\!\times\!k$ sub-matrix of $A$ by removing columns and rows of ...
11
votes
0answers
364 views

Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
2
votes
1answer
72 views

Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually, I'm not exactly looking into bipartite but my ...
3
votes
1answer
152 views

Existence of $SO(n)$-isotropic inner products which are not $O(n)$-isotropic

$\newcommand{\al}{\alpha}$ Let $M_n$ be the space of $n \times n$ real matrices. Question: For which $n$, is there an inner product on $M_n$ which satisfies: $$(*) \, \, \langle Q^TXQ,Q^TYQ ...
4
votes
2answers
242 views

Derivative of eigenvectors of a matrix with respect to its components

Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as $$ B= \sum_{i=1}^3 ...
6
votes
1answer
179 views

Fast Symbolic Linear Algebra CAS?

I am a regular user of Mathematica, Julia, and MATLAB but I am looking for something different. The problem I am trying to solve in Mathematica only requires (dense) linear algebra to specify but is ...
0
votes
1answer
128 views

Why is the Fano variety of lines on a smooth three-dimensional quadric isomorphic to $\mathbb{P}^3$?

Let $Q \subset \mathbb{P}^4$ be a smooth three-dimensional quadric over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$) and let $F$ be the Fano variety of lines on $Q$. In "Iskovskikh ...
2
votes
2answers
157 views

Intersection of Subspaces with $O(3)$

Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below. For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three ...
2
votes
1answer
152 views

programming to compute kernel quotient image of a $\mathbb{Z}$-module endomorphism

Let the integers $n\geq 2$, $k\geq 1$, $v=0$ or $1$ and $n_1,\cdots,n_k\geq 1$ such that $$ \sum_{i=1}^k n_i+v=n. $$ Define $P_a^b=0$ if both $a,b$ are odd and $P_a^b={{[a/2]}\choose {[(a+b)/2]}}$ ...
0
votes
0answers
143 views

Shortest distance on a combinatorial chess board

Consider a random $n\times n$ combinatorial $0/1$ square matrix over field $\Bbb F$ of rank $r$ with every row distinct and every column distinct as a chess board. Definition of combinatorial ...
1
vote
0answers
43 views

Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with ...
5
votes
2answers
199 views

Describing Levi factors and unipotent radicals of parabolic subgroups in classical groups

I asked this question before at Math.SE (link) but got no answer. Let $G$ be an algebraic group over an algebraically closed field $k$ of characteristic $p \geq 0$. Then any parabolic subgroup $P$ ...
4
votes
2answers
130 views

Collection of projection operators in finite dimension and algebraic techinques

Consider a set of linearly independent vectors $\{x_1,\dots,x_n\}$ in some finite-dimensional Hilbert space $H$. For any subset $S \subset [n]$, let $P_S$ be the (orthogonal) projection (operator) ...
0
votes
0answers
75 views

the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group

Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...
0
votes
1answer
86 views

Example distance metric that is not conditionally negative definite

Theorem 4.1 of this paper says that there exist distance matrices that are not conditionally negative definite (CND). How do I construct an example of a distance matrix that is not CND? Do you know an ...
0
votes
1answer
242 views

Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?

Let $\varphi_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets ...
4
votes
1answer
81 views

Kolmogorov complexity for matrices

In applications one often encounters very large matrices that barely fit in computer memory, if at all. Naturally one wishes to represent those matrices as compactly as possible. Sometimes one even ...
9
votes
1answer
185 views

Exact determinant of a circulant matrix

The wikipedia gives us a formula for the determinant of a circulant matrix. That is: $$\mathrm{det}(C) = \prod_{j=0}^{n-1} (c_0 + c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \dots + c_1\omega_j^{n-1})= ...
1
vote
1answer
107 views

Determinant of discrete Laplacian

It can easy be shown by induction that the determinant of the $(N-1)\times (N-1)$ matrix $$\begin{pmatrix} 2 & -1 & & \\ -1 & 2 & \ddots & \\ & ...
4
votes
2answers
205 views

Why are the vectors with this special structure linearly independent with high probability?

Given $a_i\in\mathbb{R}^m$ for $i=1,\ldots,2m$ with independent and identically random entries of some continuous distribution. Every choice of $m$ vectors from $\{a_1\ldots,a_{2m}\}$ is linearly ...
11
votes
1answer
431 views

An inequality for the spectral radius of matrices used by J. Bochi

I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
5
votes
2answers
196 views

Induced matrix norm less than one for matrices with spectral radius less than one

Let $A$ be a square matrix with elements in $\mathbb{R}$ or $\mathbb{C}$, $\rho\left(A\right)$ stands for the spectral radius of $A$, i.e., the maximum absolute eigenvalue of $A$; $A^{*}$ is the ...
0
votes
0answers
33 views

If $A$ is doubly stochastic and reducible. Why is $A$ permutation similar to a matrix of the form $A_1 ⊕ A_2$

If $A \in M_n$ is doubly stochastic and reducible. Why is $A$ permutation similar to a matrix of the form $A_1 ⊕ A_2$, in which both $A_1$ and $A_2$ are doubly stochastic?
1
vote
0answers
74 views

Let $M = \frac{1}{2}(A + {A^T})$ be real symmetric nonnegative matrix. Why does $\rho (A) \le {\lambda _{\max }}(M)$?

Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix $M = \frac{1}{2}(A + {A^T})$. Why does $\rho (A) \le {\lambda _{\max }}(M)$?