All Questions
Tagged with linear-algebra reference-request
318 questions
7
votes
2
answers
251
views
What methods do we have to understand the spectrum of matrices with restricted entries?
Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),
What is the largest possible spectral radius of a $...
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0
votes
1
answer
130
views
Reference for measures of commutativity needed
I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...
- 3,741
1
vote
1
answer
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 ...
- 17.9k
3
votes
3
answers
1k
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 ...
- 44.4k
5
votes
0
answers
254
views
A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
- 6,206
3
votes
2
answers
939
views
Positive definiteness of infinite tridiagonal matrices
I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \...
- 692
2
votes
0
answers
764
views
Reference: Continuity of Eigenvectors [closed]
I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, but got no answer.
For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix),...
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3
votes
0
answers
130
views
Where does this identity involving sums of Hankel-like determinants over partitions come from?
For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix $$M_f(\...
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1
vote
0
answers
90
views
Algorithms to compute the rank of a parametrized matrix [closed]
Motivated by my question on Mathematics StackExchange and by a question by Anirbit on the same site, I ask for some references on the problem of rank computation for a parametrized matrix. References ...
- 111
8
votes
0
answers
421
views
Approximate singular value decomposition in Banach spaces
I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
- 6,206
3
votes
2
answers
976
views
Positive solutions of linear systems with a diagonally dominant matrix
Given a real linear system ($\mathbf{A}\mathbf{x} = \mathbf{b}$), is there any result regarding the positiveness of the solution $\mathbf{x}^*$ considering that $\mathbf{A}$ is diagonally dominant? (...
- 342
2
votes
1
answer
336
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Bringing a (Least Squares Problem) Matrix into Block Upper-triangular Shape via Matrix-reordering
I have the problem of solving very large and very sparse least squares problems and, a bit dissatisfied with the run-times of the full-fledged QR-algorithm, I would like to bring the instances into ...
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5
votes
4
answers
1k
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 ...
- 25.3k
1
vote
0
answers
108
views
Reference request on operator matrices [closed]
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
$$Tx = \begin{pmatrix}A & B \\
C & D
\end{pmatrix}\...
- 11
4
votes
1
answer
787
views
What is a degenerate Legendre Transformation?
I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
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9
votes
1
answer
562
views
What is the total polarization of the determinant?
Let $A\in\mathfrak{gl}(\mathbb{R},n)$ be an endomorphism, and think up to conformal factors (in particular, $\Lambda^n\mathbb{R}^n$ will be the same as $\mathbb{R}$). By the total polarization $\...
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4
votes
0
answers
136
views
What do we know about the generalized eigenvalue problem involving a projector?
Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.
Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem
$$...
- 66.3k
2
votes
1
answer
520
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 $i,j \in V$, $$a_{ij} = |N_i \cap ...
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3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 12.7k
3
votes
3
answers
1k
views
Applications of rank factorization or full rank decomposition [closed]
I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r \...
- 1,897
2
votes
2
answers
654
views
Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$
I'm using the following result in a computer science paper:
Let $V$ be a submodule of $(\mathbb{Z}/q\mathbb{Z})^n$ (n-tuples with addition and multiplication mod $q$). Let
$$V^\perp = \{u \in (\...
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2
votes
3
answers
1k
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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 ...
- 1,749
3
votes
1
answer
597
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 ...
- 25.3k
4
votes
1
answer
214
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 ...
- 28.6k
31
votes
1
answer
2k
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 ...
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5
votes
1
answer
2k
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 ...
- 6,210
4
votes
3
answers
784
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 x)^*=\...
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3
votes
1
answer
389
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 ...
- 26k
3
votes
1
answer
270
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 ...
- 4,994
1
vote
1
answer
113
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} \ \mathrm{...
- 28.6k
4
votes
1
answer
255
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 ...
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29
votes
6
answers
10k
views
how to find/define eigenvectors as a continuous function of matrix?
I asked this (with background) here
https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision
but did not really get any answers. ...
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1
vote
1
answer
417
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 ...
- 54.2k
5
votes
0
answers
442
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 ...
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4
votes
0
answers
167
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}
$$
is orthogonal, $z$ ...
- 798
12
votes
2
answers
1k
views
Quadratic Farkas' Lemma?
The Farkas Lemma says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the ...
3
votes
1
answer
767
views
Linear algebra of finite abelian groups
If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
- 2,452
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 201
0
votes
0
answers
324
views
Changing basis on an extension of a free Z-module.
Consider a finite-rank free $Z$-module $Y$. Let $c: Y \times Y \rightarrow Z$ be a $Z$-bilinear form. Assume that $c(y_1, y_2) + c(y_2, y_1)$ is even, for all $y_1, y_2 \in $. Then $c$ "incarnates"...
- 13.3k
3
votes
4
answers
570
views
A polynomial homomorphism from Gl to the group of units is a power of the determinant
I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism $\mathrm{Gl}...
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5
votes
0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 12.7k
6
votes
2
answers
503
views
Unpublished work of Wielandt
Wielandt wrote a paper titled "Remarks on diagonable matrices".
According to Mathematische Werke - Mathematical Works : Linear Algebra and Analysis
by Helmut Wielandt, Hans Schneider, Bertram Huppert ...
- 11
4
votes
2
answers
1k
views
signs of eigenvalues of quadratic form
Let $A=(a_{ij})_{i,j=1}^n$ be a symmetric real matrix, $M_k:=det(a_{ij})_{1\leq i,j\leq k}$ be its minors and $M_k\ne 0$ for all $k$. Then signs of eigenvalues of $A$ are equal (up to some permutation)...
- 34.7k
4
votes
8
answers
3k
views
Theory of cones
Hi all,
Can anyone point me to some references to the theory of finitely-generated cones in euclidean space? I'd like to know in particular if there is a notion of basis/dimension/linear dependence ...
- 9,720
2
votes
1
answer
162
views
'Compute' Integral equivalence of matrices
Hi.
For a matrix $D \in \mathbb{Z}^{n \times n}$ and a symmetric, positive definite integral even matrix $S \in \mathbb{Z}^{n \times n}$ put $S[D] := D^TSD$ where the $\cdot^T$ means 'transposed'. ...
- 3,201
4
votes
1
answer
314
views
Spectral Properties of $A(I-A)^{-1}$
I am working with a class of matrices $A$ which are non-negative-definite, not symmetric, and have maximum eigenvalue less than 1. I am interested in the spectral properties of the matrix $H = A(I - ...
- 20.2k
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
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8
votes
1
answer
1k
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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 ...
- 6,871
4
votes
0
answers
189
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 ...
- 41
1
vote
1
answer
1k
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 ...
- 6,614