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

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5
votes
1answer
340 views

How many random matrices does it take to generate a matrix algebra?

Let $\mathbb{F}$ be a finite field. Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra. Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$ that one needs to take ...
9
votes
0answers
219 views

Bunnity of multilinear maps

Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
1
vote
1answer
1k views

Does this linear algebra construction based on a graph have a name, and where has it been studied?

In the paper Kochen-Specker set with seven contexts by Lisonek, Badziag, Portillo and Cabello, the following construction is used : Question : Have such constructions been used elsewhere, and if so ...
-1
votes
1answer
251 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 ...
0
votes
0answers
65 views

Degree of permutation of hypercube

Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...
2
votes
1answer
139 views

Linear independence of +/- 1 strings/vectors II

Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...
1
vote
0answers
158 views

Is there a way to simplify this apparently huge characteristic polynomial calculation?

Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let ...
1
vote
1answer
121 views

Linear independence of +/- 1 strings/vectors

Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...
8
votes
5answers
570 views

Analogue of Cayley Hamilton theorem for operators on Hilbert space

Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
5
votes
1answer
225 views

Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ ...
2
votes
3answers
365 views

A table for irreducible integral representation of finite cyclic groups

Is there such a table where the irreducible integral representations of finite cyclic groups are listed? Edited: Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
1
vote
1answer
201 views

Projectors onto the invariant subspaces of a unitary representation $U \otimes U^* \otimes U \otimes U^*$

Let $$U \mapsto U \otimes U^* \otimes U \otimes U^*$$ be a unitary representation of the unitary group $U(n)$ acting on the vector space $V$ (where $U^*$ is the complex conjugate of $U$). We can ...
44
votes
7answers
3k views

How to prove this determinant is positive?

Given the matrices $ A_i= \biggl(\begin{matrix} 0 & B_i \\ B_i^T & 0 \end{matrix} \biggr) $, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + ...
1
vote
0answers
46 views

Is there a relation between covariance matrices and real and imaginary part of eigenvectors?

Apology if my question not clear or appropriate. Consider a complex positive definite sample covariance matrix (SCM) A, generated by a band limited signal on a set of sensors which is termed as data ...
1
vote
1answer
42 views

How to characterize a linearly-constrained subspace in a projection [closed]

I hope this one is easy. Suppose I have an underdetermined, rectangular matrix $A$ and vector $b$. I want to reason about the subspace where $Ax = b$ and specifically the projection $y:= Tx$. Is there ...
4
votes
0answers
166 views

What is the complexity of intersecting two matrix algebras over a finite field?

The following question arose in a joint project with Arkadius Kalka and Adi Ben-Zvi. Let $\mathbb{F}$ be a finite field, and $M_n(\mathbb{F})$ be the $n\times n$ matrices over $\mathbb{F}$. For a ...
2
votes
2answers
160 views

constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...
0
votes
0answers
63 views

Convenient Basis Presentation of Lefschetz Decomposition

Let $V$ be an almost-complex vector space, equipped with a symplectic element $\omega \in V^{(1,1)}$. In terms of a basis $b^+_i \in V^{(1,0}$, $b^-_i \in V^{(0,1}$, does there exist a "simple" ...
5
votes
1answer
221 views

Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
2
votes
0answers
81 views

Do copairings provide dualities in derived categories?

Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...
1
vote
0answers
38 views

Lattice-isotopic essentialization of arrangements

I'm working on a problem related to $\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
0
votes
0answers
127 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
2
votes
0answers
41 views

A tensor equation related to an invariant of a diffeomorphism

Let $M$ be an $n$-dimensional differentiable manifold, $f : U \rightarrow V$ a diffeomorphism between open neighbourhoods $U$, $V$ of $M$ with $f(x)=x$ for some $x \in U$, and let $R$, $S$, $T$ be ...
1
vote
0answers
33 views

LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
3
votes
1answer
99 views

Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed

The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed: I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n ...
9
votes
2answers
293 views

Determinant of a checkerboard Hankel matrix with Catalan numbers

My goal is to compute \begin{equation} I = \det \left(\mathbf{I} + \mathbf{A}\right) \end{equation} where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers: $$ \left\{ ...
1
vote
0answers
80 views

Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$. Define a sequence of banded ...
2
votes
1answer
113 views

Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem

Dr. Tao in his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality. Specifically, I am looking for proof of Eq. (13) where Dr. Tao ...
3
votes
2answers
156 views

Is the exterior power of a primitive matrix still primitive?

the question is already in the title. Here some more details. I have a primitive matrix $M$ (primitive means $\exists k\geq 0$ such that $M^k > 0$). I take exterior powers $\wedge^n M$ and I would ...
2
votes
1answer
232 views

Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent

I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...
8
votes
1answer
295 views

Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
1
vote
1answer
69 views

Self adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator H acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
14
votes
1answer
851 views

How many values determine a norm?

It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely. How many values do we need to specify in order to ...
22
votes
4answers
1k views

Dividing by two in the category of vector spaces

Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$? Here "naturally" means "in an $GL(V) \times ...
0
votes
2answers
70 views

Symmetric matrix from a nonsymmetricc matrix

Basically this is a part of a long algorithm to calculate some matrix properties. Given an upper triangular square matrix R, how can I find an orthonormal matrix W (possibly iteratively) such that WR ...
1
vote
0answers
110 views

Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$. B is any $n \times n$ n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: ...
2
votes
0answers
55 views

Smallest Singular Value of a Random Matrix with Dependent Entries

Overview I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to ...
1
vote
0answers
69 views

Complex Hessian Signature

It' all, simply, about the signature of a matrix. Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
3
votes
1answer
170 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
1
vote
1answer
183 views

About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...
1
vote
2answers
110 views

Solving $Ax=e_k$ for standard basis vector $e_k$, sparse $A$

Given a sparse matrix $A \in \mathbb{R}^{n \times m}$, are there any efficient methods for determining whether there exists an $x \in \mathbb{R}^m$ such that $Ax=e_k$, the $k^{th}$ standard basis ...
1
vote
0answers
30 views

Volume under the intersection of scaled simplices

This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space: $\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; k = 1 ...
0
votes
1answer
107 views

Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices. Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...
1
vote
0answers
79 views

Notions of consistency / heterogeneity in sets of vector values?

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
1
vote
1answer
109 views

Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
0
votes
0answers
99 views

Hadamard / matrix product adjoint

First of all I would like to thank everyone over here at mathoverflow for their amazing generosity and help (for both pros and newbies like myself). I apologize if this question seems dumb; I'm a new ...
1
vote
1answer
112 views

Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...
6
votes
5answers
381 views

The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$. Is there any ...
1
vote
0answers
92 views

Showing a wedge product is nonzero

Let $V$ be a complex vector space of dimension $n$, equipped with a Hermitian inner product whose Kahler form we denote by $\omega$. Let's set $P = \bigwedge^{2p} V^*$ and $Q = \bigwedge^{2q} V^*$ for ...
1
vote
2answers
99 views

Expectation of Gaussian random vector & arbitrary function thereof?

I saw in a paper (https://www.princeton.edu/~wbialek/rome/refs/bialek+ruyter_05.pdf Eq.37) the following identity: where the <.> operator refers to a population average. No source or ...