All Questions
Tagged with linear-algebra reference-request
318 questions
27
votes
6
answers
6k
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 ...
- 443
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
$$...
- 325
3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 6,962
4
votes
1
answer
630
views
Can sparse matrices satisfy the Null Space Property?
Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if
$$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus \{...
- 837
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 \...
- 357
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
$$V^\perp = \{u \in (\...
- 123
3
votes
1
answer
597
views
Has anybody seen my missing lemma?
I think I have a proof of the following elementary lemma (although I only need the case in which the two flags are "in general position", i.e., $F^d \cap G^i$ is minimal given the dimensions of the ...
- 7,338
4
votes
1
answer
214
views
The d-dimensional matrix with columns (1,0,0…), (1/2,1/2,0,…), (1/3,1/3,1/3,0,…),…, (1/d,1/d,…,1/d)
During the course of physics research on nonequilibirum statistical mechanics involving the theory of majorization, I have come across a linear transformation on a d-dimensional vector space that I ...
- 41
4
votes
3
answers
784
views
A textbook on linear algebra where involutions on linear spaces are considered
Let us call an involution on a complex linear space $X$ an arbitrary $\mathbb R$-linear map $x\in X\mapsto x^*\in X$ that satisfies the following identities:
$$
x^{**}=x,\qquad (\lambda\cdot x)^*=\...
- 7,426
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 ...
- 303
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 ...
- 63
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 ...
- 7,633
3
votes
1
answer
3k
views
Diagonalize the simultaneous matrices and its background [closed]
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 $F$ which ...
- 8,071
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{...
- 3,323
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 ...
- 10.5k
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. ...
- 2,600
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 ...
- 4,784
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 ...
- 23k
5
votes
2
answers
2k
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 ...
- 191
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 ...
- 133
36
votes
2
answers
32k
views
Eigenvalues of the 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 $BA$...
- 2,600
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 ...
- 4,010
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
2
votes
3
answers
1k
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 ...
- 355
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 ...
- 19.7k
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}...
- 4,902
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
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 922
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 ...
- 2,600
1
vote
0
answers
229
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 ...
- 5,898
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 ...
- 2,829
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)...
- 109k
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'. ...
- 303
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 - ...
- 8,512
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_{...
- 1,222
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 ...
- 757
8
votes
1
answer
1k
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 ...
- 4,343
4
votes
3
answers
755
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 ...
- 406
5
votes
3
answers
1k
views
adjoint of multiplication operator in a commutative algebra
Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\...
- 6,614
1
vote
0
answers
201
views
What is the MP pseudoinverse's role in statistical learning and Self-Organizing Maps?
During a discussion in our lab last month, a professor mentioned to me that the behavior of Self-Organizing Maps can be described in terms of repeated applications of the Moore-Penrose psuedoinverse, ...
- 113
9
votes
1
answer
904
views
Reference for a formula expressing the characteristic polynomial of a sum of endomorphisms
Let $R$ be a ring, $A$ a (not necessarily commutative) $R$-algebra and $M$ a (left) A-module which is free of finite rank as an $R$-module. If $a\in A$ then multiplication with $a$ on $M$ is an $R$-...
- 1,645
1
vote
2
answers
763
views
A basis for $\mathbb{Q_p}$ as a vector space over $\mathbb{Q}$
I was looking for a reference that illustrates a $\mathbb{Q}$-vector space basis for the field of p-adic numbers under the following action. Given a rational number $q$. write, $q=\frac{m}{n}$ where $...
3
votes
1
answer
621
views
Largest eigenvalue of a periodic Jacobi matrix
There is a vast literature on Jacobi matrices, I just don't know where to start looking. I'm interested in estimating the largest eigenvalue of the $n\times n$ periodic Jacobi matrix $D+P+P^{-1}$, ...
- 11.1k
1
vote
2
answers
3k
views
Fast algorithms for computing nullspace of a positive semidefinite matrix over Z
Let $A \in \mathbb{Z}^{n \times n}$ be a positive semidefinite sparse matrix. I am looking for asymptotically fast algorithms for computing the nullspace basis of $A$ (or just random elements in the ...
- 236
8
votes
1
answer
248
views
Operator compression preserving lowest energy eigenspace.
I have a large ($10^6$ by $10^6$) sparse ($0.4$% nonzero) hermitian matrix $H$ arising from the discretization of an elliptic PDE. I would like to approximate $H$ with a smaller matrix $H'$ in such a ...
- 817
12
votes
3
answers
2k
views
Representability of matroids over $\mathbb R$
Let $M$ be a matroid, for example viewed as being given by a finite set $X$ and a rank function $d : P(X) \to {\mathbb N}$ such that
1) $d(\varnothing)=0$, $d(\lbrace x \rbrace)=1$, for all $x \in X$,...
- 25.5k
6
votes
0
answers
998
views
Generalized Courant-Fischer theorem
Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
- 475
6
votes
3
answers
590
views
Zariski-closed subsemigroups of SL_n(C) are groups
I would like to show that any Zariski-closed subsemigroup of $SL_n(\mathbb{C})$ is a group. If I understand correctly, this is consequence 1.2.A of http://www.heldermann-verlag.de/jlt/jlt03/BOSLAT.PDF ...
- 892
1
vote
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922