5
votes
1answer
73 views

What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are individual eigenvectors ...
1
vote
0answers
69 views

Reference request on operator matrices [on hold]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that $$Tx = \begin{pmatrix}A & B \\ C & D ...
5
votes
0answers
269 views

Reference request: Book of Linear algebra from categorical point of view

Is there any book of Linear algebra in the modern language of Category theory? I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are ...
7
votes
3answers
360 views

Diagonalization via the Toda flow

inAccording to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...
3
votes
1answer
121 views

What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
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 ...
7
votes
1answer
277 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 ...
0
votes
0answers
50 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
82 views

Existence of parametrizations of rational orthogonal matrices

I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this? ...
1
vote
1answer
99 views

Neighborhood overlap matrix for a bipartite graph

Let $G$ be an undirected, simple, bipartite graph with parts $V$ (having $n$ vertices) and $W$ (having $m$ vertices). Define the following $n$-by-$n$ matrix: for any $i,j \in V$, $$a_{ij} = |N_i \cap ...
15
votes
5answers
2k views

Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
3
votes
0answers
74 views

What do we know about the generalized eigenvalue problem involving a projector?

Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$. Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem ...
3
votes
1answer
109 views

The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$. This situation has many different names: $G$ is called the cone or the ...
2
votes
1answer
198 views

Can sparse matrices satisfy the Null Space Property?

Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if $$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus ...
3
votes
3answers
177 views

Applications of rank factorization or full rank decomposition [closed]

I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r ...
2
votes
2answers
216 views

Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$

I'm using the following result in a computer science paper: Let $V$ be a submodule of $(\mathbb{Z}/q\mathbb{Z})^n$ (n-tuples with addition and multiplication mod $q$). Let $$V^\perp = \{u \in ...
0
votes
0answers
96 views

Perturbation of spectrum and eigenspaces

Let $A \in \mathbb{C}^{n \times n}$ be an $n \times n$ matrix. Consider the rank-$1$ perturbation $A'$ of $A$ given by replacing a column $v$ of $A$ by $\alpha \cdot v$, where $\alpha \in [0, 1)$. Can ...
3
votes
1answer
508 views

Has anybody seen my missing lemma?

I think I have a proof of the following elementary lemma (although I only need the case in which the two flags are "in general position", i.e., $F^d \cap G^i$ is minimal given the dimensions of the ...
4
votes
1answer
207 views

The d-dimensional matrix with columns (1,0,0…), (1/2,1/2,0,…), (1/3,1/3,1/3,0,…),…, (1/d,1/d,…,1/d)

During the course of physics research on nonequilibirum statistical mechanics involving the theory of majorization, I have come across a linear transformation on a d-dimensional vector space that I ...
2
votes
3answers
373 views

A textbook on linear algebra where involutions on linear spaces are considered

Let us call an involution on a complex linear space $X$ an arbitrary $\mathbb R$-linear map $x\in X\mapsto x^*\in X$ that satisfies the following identities: $$ x^{**}=x,\qquad (\lambda\cdot ...
4
votes
1answer
341 views

Rank of a 0-1-matrix

Suppose $K$ is a field of characteristic $0$. Let $M \in K^{n \times m}$ be a matrix such that every entry of $M$ is either $0$ or $1$. About this matrix, I know further that each sum over a column ...
2
votes
1answer
228 views

Galois deformations with Panchiskin condition

Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...
3
votes
1answer
156 views

What is the name of this measure of matrix “degenerateness”

Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called? If the vectors are columns in a matrix $\Phi$, then ...
3
votes
1answer
593 views

Diagonalize the simultaneous matrices and its background

For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$: Question1:Is there always a nonsingular matrix $P$ over the same field $F$ ...
1
vote
1answer
82 views

Expected rank - computable approximations

I'm interested in finding the expected rank of some random matrix $A$ (I don't want to specify its distribution right now, since my question makes sense in general). Computing $\mathbb{E} \ ...
3
votes
1answer
184 views

On the divisibility of the special linear group of degree $n$ over an algebraically closed field

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But ...
16
votes
6answers
1k views

how to find/define eigenvectors as a continuous function of matrix?

I asked this (with background) here http://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision but did not really get any answers. ...
1
vote
1answer
188 views

Decomposition of Matrix to its sub-matrix with constant rank

When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the ...
3
votes
0answers
119 views

Reduction of size of orthogonal matrices.

While experimenting with orthogonal vectors I've noticed the following transformation: If $$ A = \begin{bmatrix}z & r \cr c & B\end{bmatrix} $$ is orthogonal, $z$ ...
11
votes
2answers
520 views

Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies yet another linear inequality, then this last inequality can be obtained by taking a positive linear combination of the ...
3
votes
2answers
241 views

Bounding the minimal maximum norm of a solution of a linear system.

I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...
2
votes
1answer
494 views

Linear algebra of finite abelian groups

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
6
votes
1answer
3k views

Eigenvalues of product of two symmetric matrices

This is mostly a reference request, as this must be well known! Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or ...
5
votes
0answers
403 views

A reference on semisimple linear algebra

Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings? In fact this question is a ...
0
votes
0answers
164 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 $c$ ...
2
votes
3answers
400 views

On certain decomposition of unitary symmetric matrices

This is by any means elementary, but since I have asked this question on Stark Exchange but received no satisfactory answers I decide to post it here. It is well known that a symmetric matrix over ...
26
votes
1answer
990 views

solving linear equations made difficult

(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.) I saw this amusing derivation ...
2
votes
4answers
319 views

A polynomial homomorphism from Gl to the group of units is a power of the determinant

I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism ...
5
votes
0answers
125 views

reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then $$ \|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2). $$ ...
3
votes
4answers
773 views

determinants and polynomials in matrices

Muirhead (1982, "Aspects of Multivariate Statistical Theory") references on page 59 a result (from MacDuffee, 1943, chap 3, "Vectors and Matrices") a book I cannot find): " The only polynomials in ...
1
vote
0answers
113 views

Counting equivalence classes in the transitive closure of two equivalence relations

Let $X$ be a finite set, and let $P_i$ and $Q_j$ be two partitions of $X$: $$\bigsqcup_i P_i = \bigsqcup_j Q_j = X.$$ The finest partition which is nevertheless coarser than both $P$ and $Q$ is ...
5
votes
2answers
433 views

Unpublished work of Wielandt

Wielandt wrote a paper titled "Remarks on diagonable matrices". According to Mathematische Werke - Mathematical Works : Linear Algebra and Analysis by Helmut Wielandt, Hans Schneider, Bertram Huppert ...
4
votes
2answers
427 views

signs of eigenvalues of quadratic form

Let $A=(a_{ij})_{i,j=1}^n$ be a symmetric real matrix, $M_k:=det(a_{ij})_{1\leq i,j\leq k}$ be its minors and $M_k\ne 0$ for all $k$. Then signs of eigenvalues of $A$ are equal (up to some ...
2
votes
1answer
139 views

'Compute' Integral equivalence of matrices

Hi. For a matrix $D \in \mathbb{Z}^{n \times n}$ and a symmetric, positive definite integral even matrix $S \in \mathbb{Z}^{n \times n}$ put $S[D] := D^TSD$ where the $\cdot^T$ means 'transposed'. ...
4
votes
1answer
255 views

Spectral Properties of $A(I-A)^{-1}$

I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...
1
vote
1answer
192 views

references for families of conditionaly negative definite matrices

We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have $$ ...
4
votes
0answers
139 views

Slices of Simplices that are Simplices, Reference?

I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere. Let $A$ be an $l\times n$ matrix with ...
1
vote
1answer
535 views

Characterizing the set of self-orthogonal complex vectors

Let $v\in \mathbb{C}^n$ be an $n$ dimensional complex vector. Define the non-standard bilinear form $\left< u,v \right> = u^T v$ (the usual inner product except without the conjugation). What ...
7
votes
1answer
563 views

Calculating a curvature tensor by polarization

I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
3
votes
3answers
476 views

Is this statement about the real edge space of a graph known or trivial?

The statement is: ($u$ is a fixed node in a fixed graph $G$) $G$ is 3-connected if and only if the set of u-cycles span $\mathbb{R}^{E(G)}$. A u-cycle is a simple (no vertex repetitions) cycle in G ...