All Questions
Tagged with linear-algebra reference-request
318 questions
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 ...
- 37.7k
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, ...
CommunityBot
- 1
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 $R$-...
- 361
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 $...
- 41.1k
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 $D+P+P^{-1}$, ...
- 3,343
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 ...
- 188k
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 ...
- 379
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 ...
- 33.8k
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
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
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 10.7k
4
votes
1
answer
1k
views
Integer vectors in the kernel of an integer matrix
Let $A$ be a non-zero symmetric $n \times n$-matrix with integer entries and suppose that $\det(A) =0$.
Question: How long is the shortest non-zero integer vector in the kernel of $A$?
Example: If ...
CommunityBot
- 1
4
votes
1
answer
254
views
Embedding into Permutation Representation
Let $\rho$ be irreducible representation of group $G$.
How one can characterize all subgroups $H< G$ such that $\rho$ can be embedded into permutation representation $F^X$, where $X=G/H$.
- 52.9k
2
votes
0
answers
1k
views
Good sources for linear algebra for convex optimization and graph analysis?
What are some good sources for linear algebra for convex optimization and graph analysis?
In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses (...
- 2,383
8
votes
2
answers
3k
views
Centralizers in GL(n,p)
There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is ...
- 10.7k
4
votes
0
answers
352
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"Cholesky decomposition" X=YY* for p-adic matrices?
Let $E/F$ be a quadratic extension of $p$-adic fields. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $...
- 932
7
votes
1
answer
727
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Reference for Tate vector spaces
... aka locally linear compact vector spaces. The one reference I know is http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov3-10(CentExt).pdf. Does anyone know another good reference?
- 45.7k
2
votes
2
answers
3k
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Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 33.8k