All Questions
6,177 questions
0
votes
2
answers
415
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Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$
A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space $l^...
- 25.5k
4
votes
2
answers
322
views
Algebra with elements x, y such that r(x)=r(y) for all finite-dimensional modules r
I'm interested in finding an algebra with elements x,y which are identified by every finite-dimensional module. I'm primarily interested in the case that everything is over the complex field, but ...
- 38.6k
6
votes
3
answers
2k
views
Finite subgroup of $Gl(n,\mathbb Z)$ and congruences
Suppose we have an invertible matrix q in a finite subgroup $Q$ of
$Gl(n,\mathbb Z)$, the group of all invertible integer matrices. Now I want to
find all $x\; mod\; \mathbb Z^n$ for which
$(q+q^2+q^...
- 16.2k
1
vote
1
answer
459
views
Looking for name of a famous matrix
Let $A_n$ be the $n\times n$ matrix whose $(i,j)$-element is $1/(i+j-1)$. This is a famous matrix in linear algebra and has some nice properties (like, its inverse is integral).
Does anybody remember ...
- 47.3k
4
votes
1
answer
354
views
Does the weak approximation theorem hold for general topological fields?
The weak approximation theorem states that given a field $F$ and nontrivial inequivalent absolute values $|\cdot|_1,\ldots,|\cdot|_n,$ and letting $F_i$ denote $F$ with the topology from $|\cdot|_i$, ...
CommunityBot
- 1
4
votes
1
answer
298
views
Is there a standard name for the intersection of all maximal linearly independent subsets of a given set in a vector space?
The title more or less says it all.... Let $V$ be a vector space (over your favorite field; $V$ not necessarily finite dimensional), and let $S$ be a subset of $V$. A maximal linearly independent ...
- 44.4k
5
votes
2
answers
3k
views
Complex Eigenvalues of Directed Graphs
I have been computing eigenvalues of adjacency matrices for several directed (not necessarily strongly connected) graphs and one remarkable property seemed to hold (each graph that I have examined ...
- 12.1k
0
votes
1
answer
189
views
Can extremal matrices of subcones of psd matrices have low rank?
Let $S$ be the cone of positive semidefinite symmetric real matrices of size $n\times n$. The cone $S$ spans a $d:=n(n+1)/2$ dimensional vector space.
Let $C\subset S$ be a subcone formed by ...
- 54.2k
1
vote
0
answers
606
views
how to prove uniqueness of matrix polynomial and its eigendecomposition
Hello, all!
Let $\underset{l \times l}{A(x)}$ be square polynomial matrix over $GF(q)[x]$, where $q$ is a prime power. Let $x_i \in GF(q^m)$ ($x_i \not= 0$) be eigenvalue of $A(x)$: $det(A(x_i)) = 0$....
CommunityBot
- 1
6
votes
1
answer
448
views
Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices
Dear community,
I have the following combinatorial question which I will explain in short first and then with some more detail. At the end you will find a very simple example.
Short version
Le $A \...
- 4,705
8
votes
2
answers
471
views
Multiplying functions on the unit square as generalized matrices
Consider the $\mathbb{R}$-vector space of sufficiently nice real-valued functions on the unit square $I^2$, where "sufficiently nice" could be taken to mean any one of a number of things - say ...
- 3,908
2
votes
1
answer
228
views
Computing a generating set of the kernel of a module
Crossposted from math.stackexchange, since I'm not getting any answer and I think the question is suitable here.
Given a generating set of a $\mathbb{Z_k}$-module $M \subseteq {\mathbb{Z}_k}^n$, is ...
CommunityBot
- 1
2
votes
2
answers
164
views
Looking for a simple proof that the generalized disc is bounded
So let us define the generalized disc of degree $n$ as
$$
\mathbb{D}_n:=\{w\in M_{n\times n}(\mathbb{C}):w=w^t, I_n-w\overline{w}>0\}.
$$
For a Hermitian matrix $A$, the notation $A>0$ means ...
- 7,273
1
vote
1
answer
409
views
Encoding information about submatrix determinants
$M$ is an $n\times n$ matrix. Consider the submatrices $M(P;Q)$ formed from $P$ rows and $Q$ columns of $M$ where $P$ and $Q$ are disjoint indices.
Is there some way to encode the various ...
- 21.5k
13
votes
2
answers
1k
views
Seeking proof for linear algebra constraint problem.
Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all ...
- 9,486
-2
votes
1
answer
327
views
A Matrix equation
Let $A$ and $B$ be two $n \times n$ full-rank matrices.
Let $XAY = B$ be the given equation where $X$ and $Y$ are unknown $n \times n$ matrices. We know that $Vec(B) = (Y^{T} \otimes X)Vec(A)$. Under ...
- 8,559
10
votes
1
answer
2k
views
Multilinear generalization of Cauchy-Schwarz inequality
Let $V$ be a real vector space, and let $(\cdot,\cdot;\cdot,\cdot) : V^4 \to \mathbb{R}$ be a multilinear form with the following properties:
$(x,y;z,w) = (y,x;z,w) = (x,y;w,z)$ (symmetry in the ...
- 29.8k
3
votes
0
answers
432
views
Explicit expression for determinant
I would like to get an explicit expression for the determinant of a Jacobian
$$J_{ij} = \delta_{ij}\left(\sigma^{2}+\sum_{k}w_{ik}y_{k}^{2}\right)-w_{ij}y_{i}y_{j},$$
where $i,j = 1,...,n$, $w_{ij}\...
- 256
7
votes
1
answer
505
views
Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently
What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$?
Do we have to calculate $A^{-1}b$, or is this not necessary?
edit: I forgot to mention that A is symmetric and ...
- 461
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
6
votes
2
answers
5k
views
Periodic matrices
A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix.
Can we characterize the periodic matrices in $\mathcal{M}_n(\mathbb{Z})$?
If we replace $\mathbb{...
CommunityBot
- 1
3
votes
1
answer
362
views
Eigenvalues of certain positive matrices
For a matrix $ Q = (q_{ij}) \in GL_n(\mathbb{C}) $ let
$ \overline{Q} = (\overline{q_{ij}}) $ be the matrix obtained by entry-wise complex
conjugation (equivalently, $ \overline{Q} $ is the ...
- 3,217
0
votes
1
answer
325
views
Power of an order relation
Let there be > included in AxB as a binary relation.
What does (x)>^2(y) mean? What is the meaning of an order relation raised to a power?
My first tought was that >^2 = >x> which is a cartesian ...
- 236k
3
votes
0
answers
384
views
Systems of linear octonionic equations
Is there theory of determinants, rank of matrices and systems of linear equations with octonionic coefficients? Does anybody could indicate references? I want to know mainly does there exist a ...
- 2,135
8
votes
1
answer
1k
views
Spectra of a Symmetric Toeplitz Operator
For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form
$\begin{bmatrix}
...
- 21
9
votes
3
answers
4k
views
2-norm of the upper triangular "all-ones" matrix
Let $M_n$ be the $n\times n$ matrix
$$
(M_n)_{ij}=\begin{cases}1 & i\leq j,\\\\ 0 &i>j.\end{cases}
$$
Is there around an explicit expression or at least an asymptotic for $\left\Vert M_n \...
- 44.4k
4
votes
0
answers
382
views
Pseudoinverse of column submatrix, from pseudoinverse of entire matrix.
Hello,
I am working on a numerical method for the least-squares solution of a linear system. I know that I can approximate the solution to $Ax=b$ with $x=A^+b$, where $A^+$ is the Moore-Penrose ...
- 41
2
votes
1
answer
320
views
Update to SVD by changing 2 row vectors
Suppose I have a matrix in the form
$\begin{bmatrix}
a_1 \\
B \\
c_1
\end{bmatrix}
$
(The blocks of this matrix should be vertically stacked...don't know why the latex is wrong)
and its SVD is X. ...
16
votes
3
answers
2k
views
Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why?
Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems.
The ...
- 63.1k
12
votes
1
answer
3k
views
Matrix inversion lemma with pseudoinverses
The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own.
Suppose we pick $n$ values $x_1,\...
1
vote
0
answers
1k
views
Maximum of Tr$(ABA')$
Let $B$ be a fixed symmetric $M\times M$ matrix over the reals.
Let $A$ be an arbitrary $N\times M$ matrix over the reals.
I want to consider the problem of finding the extremal value of $\...
- 39.1k
4
votes
1
answer
535
views
A Question on Koszul duality and $B(\infty)$ structures on $HH^*$
The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type.
There is an equivalence of Gerstenhaber ...
CommunityBot
- 1
0
votes
1
answer
222
views
Is the following DNN matrix CP?
Is the following Doubly Non-negative matrix Completely Positive:
$\frac{1}{6}\begin{bmatrix} 2 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\0 & 2 & 0 & 1 & 0 &...
- 13k
1
vote
1
answer
158
views
On the boundedness of linear representations of formal power series of languages.
Let $\Sigma$ be a finite non-empty set of symbols (i.e. an alphabet). Fix $\pi, \eta\in\mathbb{R}^{1\times m}$ and for every $\sigma\in\Sigma $ fix $A(\sigma)\in\mathbb{R}^{m\times m}$.
We also ...
- 424
3
votes
2
answers
2k
views
trace norm inequality for positive matrices
If $A, B$ are positive $n \times n$ complex matrices, $n$ some integer, then obviously \begin{equation*} \|ABA\|_\text{tr} = tr(ABA) = tr(A^2 B). \end{equation*}
But can we say there is a constant $...
- 9,486
6
votes
2
answers
740
views
Is there a name for this type of matrix? (Reference Request)
I am working on a problem were I encounter matrices of the form
$X = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$
I am aware of Cauchy matrices, which have the form
$X = \begin{bmatrix}\frac{...
CommunityBot
- 1
13
votes
1
answer
13k
views
Eigenvalues of submatrices
I am interested in results on the eigenvalues of submatrices.
Given a symmetric and positive-semidefinite matrix $M$, denote the submatrix obtained by deleting the ith column and jth row as $[M]_{ji}$...
- 21
0
votes
1
answer
551
views
Surface of the cut of an ellipsoid / Marginal density of a multivariate normal over an affine space
So I'm trying to get the marginal density of a multivariate normal over an affine space
if $A$ is a matrix in $\mathbb{R}^p \times \mathbb{R}^n$ for $p < n$ and $B \in \mathbb{R}^n$, $\Sigma$ is a ...
CommunityBot
- 1
8
votes
2
answers
1k
views
Solving the equation $xax=b$ in a C*-algebra.
Let $a, b\in A_+$ be positive elements of some C*-algebra $A$.
Assume furthermore that $a$ is invertible.
Is it true that
$$
\exists! x\in A_+\quad:\quad xax=b\quad ?
$$
Already in the case $A=M_2(\...
- 43.2k
5
votes
1
answer
2k
views
annihilator/common left multiple of matrix polynomials
Let $A_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. ...
CommunityBot
- 1
5
votes
1
answer
1k
views
Ask some matrix eigenvalue inequalities.
Let $ \begin{bmatrix}
A& B \\\\ B^* &C
\end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$.
Are the following plausible inequalities true? I have seen a lot of ...
- 1,959
16
votes
3
answers
1k
views
symmetric integer matrices
Suppose I have a symmetric positive definite matrix $M$ with integer entries. I want to decide whether $M = A A^t,$ with $A$ likewise integral. I assume that decision problem is NP-complete, as is the ...
CommunityBot
- 1
4
votes
2
answers
3k
views
modified gram schmidt...
So I understand that the effective formula for the orthogonal basis of a matrix is the same in both modified and classical Gram Schmidt algorithm. Can someone explain whats the numerical instability ...
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
10
votes
1
answer
5k
views
Eigendecomposition after multiplying by diagonal matrix
Hello,
If we possess the eigendecomposition of a positive definite matrix: $X = U \Sigma U^T$, is there an efficient way to compute the eigendecomposition of $D X D$ where $D$ is a diagonal matrix?
- 156k
2
votes
1
answer
200
views
Planar Graphs and Skew Binary Spaces
Let $G$ be a planar triangulation on $3m$ edges and $m+2$ vertices. Let $A$ be the binary matrix obtained from the incidence matrix of $G$ by deleting a row (equivalently we require the rows of $A$ to ...
- 12.1k
8
votes
1
answer
646
views
Projecting the unit cube onto a (very special) subspace
Let $n>1$ be an integer, and $a>1$ a real number. Consider the subspace $L<R^{2^n}$ generated by the $n$ possible tensor products of the $n-1$ copies of the vector $(1,a)$ and one copy of $(a,...
- 23k
0
votes
1
answer
503
views
When are operators extended by linearity bounded?
Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
- 73.2k
1
vote
2
answers
762
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
2
votes
1
answer
3k
views
Subtract diagonal terms from the matrix to make it negative semi-definite
I'm reading one paper and on page 36 (48 in the pdf) it says:
Let d(s, i) be the (positive) diagonal terms that need to be subtracted from the matrix to make it negative semi-definite...
...
- 156k