All Questions
6,292 questions
8
votes
1
answer
911
views
A Problem on Linear Algebra
I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity:
Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
- 153
9
votes
3
answers
7k
views
Set of Positive Definite matrices with determinant > 1 forms a convex set
While reading a paper An Arithmetic Proof of John’s Ellipsoid Theorem by Gruber and Schuster, I have a question on their proof.
Consider an $n\times n$ real symmetric and positive definite matrix $\...
2
votes
1
answer
1k
views
Updating $LU$ decomposition after adding a sparse matrix
How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing ...
- 2,205
4
votes
2
answers
1k
views
Diagonalization of quaternion matrices
I would like to diagonalize a very large matrix that has the same property as quaternion matrices (in a sense that the matrix can be written as a linear combination of quaternions):
$$
H = \left(\...
2
votes
0
answers
130
views
A - B is semidefinite, what the relationship about their eigenvalues? [closed]
$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...
- 25
19
votes
4
answers
7k
views
Sherman-Morrison type formula for Moore-Penrose pseudoinverse
Given an $n\times n$ invertible matrix $\mathbf A$ and two column vectors $\mathbf u$, $\mathbf v\in\mathbb R^n$, suppose that $1 + {\mathbf v}^T {\mathbf A}^{-1}\mathbf u \neq 0$.
Then the Sherman-...
0
votes
0
answers
704
views
expected matrix inverse of circulant plus diagonal matrix with chi-square variables
Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$.
Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...
- 96
7
votes
3
answers
2k
views
Optimization problem on trace of rotated positive definite matrices
Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R \,\...
- 362
2
votes
1
answer
359
views
Dimension independent computational complexity of singular value decomposition
Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...
- 45
3
votes
3
answers
611
views
On MDS code property
Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code?
Is there an easy test? If so, could someone provide ...
- 13.9k
3
votes
0
answers
770
views
Kullback-Leibler Divergence of Stationary Distributions of Markov chains
Consider two finite Markov chains on the same state space, both assumed to be irreducible, with transition matrices $P$ and $Q$ and associated stationary distributions $\pi$ and $\tilde \pi$. Is it ...
- 418
0
votes
1
answer
134
views
Is the Hodge Map Unitary?
Let $(V,< \cdot, \cdot >)$ be an inner product space over a field ${K}$. As usual, we can extend $< \cdot, \cdot >$ to a mapping on the exterior algebra of $V$ using the usual matrix ...
4
votes
0
answers
990
views
Probability distribution function for singular value sum of Gaussian random matrix
Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition (...
6
votes
1
answer
371
views
Separating the spectrum of a Hermitian matrix
Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated.
Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each
...
- 445
2
votes
0
answers
773
views
Expected mean square error of an estimation problem
Let R be an $(N+1)\times (N+1)$ Toeplitz matrix. I would be mostly interested in the case of $N\to \infty$. Let $x$ be a complex Gaussian random $N\times 1$ vector with mean zero and covariance matrix ...
- 96
3
votes
1
answer
303
views
Maps between general linear group that can be extended to functor
this is just a basic linear algebra question, which I do not have a idea.
Suppose that we have a group homomorphism $\phi:GL_n(\mathbb{R})\rightarrow GL_m(\mathbb{R})$. Is there always a functor $F:...
- 31
2
votes
0
answers
137
views
equalizing diagonals of a matrix inverse
Let $Z$ be a square matrix; denote by $\text{diag}_k(Z)$ the matrix containing the center $2k+1$ diagonals of $Z$; i.e., if $T=\text{diag}_k(Z)$, then
$$T_{ij}=Z_{ij}, \;\mathrm{if}\; |i-j|\leq k, \...
- 96
7
votes
2
answers
2k
views
Powers of singular matrices and pairs of identical rows
Let $A$ be a square real or complex matrix. We’ll call $A$ special if among its rows (or among its columns) there are two identical ones, different from the zero vector, (Added:) and if it has no ...
- 13.4k
12
votes
2
answers
3k
views
How to project a vector onto a very large, non-orthogonal subspace
I have a difficult problem.
I have a very large, non-orthogonal matrix $A$ and need to project the vector $y$ onto the subspace spanning the columns of $A$. If this were a small matrix, I would use ...
- 223
6
votes
3
answers
2k
views
Are all (possibly infinite dimensional) irreducible representations of a commutative algebra one-dimensional?
If $A$ is a commutative algebra over an algebraically closed field $k$, and $\rho:A \rightarrow End(V)$ is an irreducible representation of $A$ (where, a priori, $V$ may be infinite dimensional), can ...
- 922
4
votes
1
answer
260
views
The Maslov triple product is alternating in its entries
Let $(V,\omega)$ be a $2g$-dimensional symplectic vector space. I'm trying to understand the Maslov triple product. I know that it can be defined in a variety of ways, but for the applications I'm ...
- 41
5
votes
1
answer
908
views
Can one characterize the category of finite-dimensional vector spaces? [duplicate]
Let $K$ be a field. Does the category of finitely generated $K$-modules have a nice characterization, for example as the unique abelian category satisfying a certain simple condition? For example, we ...
- 3,761
4
votes
0
answers
168
views
Level sets of linear combinations of Gaussians
I am trying to work out whether level sets of linear combinations of Gaussian functions are unique.
For a given integer $n\ge 1$, fix $n$ points $x_i\in\mathbb{R}^d$ and $\sigma>0$. Let $\mathcal{...
- 51
4
votes
1
answer
398
views
Distribution of the spectrum of a perturbed matrix
Let $A$ be an $n\times n$ Hermitian matrix,
with well-separated eigenvalues $\lambda_1 > \lambda_2 ... > \lambda_n$,
with $|\lambda_i-\lambda_j|>\epsilon$, for all $i \neq j$.
Let $G$ be a ...
- 445
2
votes
2
answers
170
views
Given a subdomain of GL(n), when is the map from matrices to their matrices of eigenvectors a diffeomorphism?
I'm wondering if there are any general conditions on a subdomain of $GL(n)$, which would guarantee that the map from a matrix to its matrix of eigenvectors is a diffeomorphism.
For example, given a ...
- 173
15
votes
3
answers
24k
views
How to solve this quadratic matrix equation?
I would like to solve for $X$ in the matrix equation
$$
XCX + AX = I
$$
where all the matrices are $n\times n$, have real components, $X$ is positive semidefinite and $C$ is symmetric. My (possibly ...
- 395
6
votes
4
answers
884
views
A name for matrices with only simple eigenvalues?
I am constantly working with hermitian matrices without multiplicity in their spectrum. Since this hypothesis appear in several important problems, for instance perturbation theory, I looked in the ...
- 452
2
votes
1
answer
358
views
positive semidefinite matrix condition
There is a great work of Alizadeh that in section 4 speaks about Minimizing sum of the first few(k-largest) eigenvalues of a symmetric matrix. Instead of a symmetric model we use the weighted ...
- 161
7
votes
0
answers
384
views
Concept of eigenvector restricted to nonnegative entries
Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem
$\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...
- 373
4
votes
1
answer
1k
views
How to solve a matrix equation with both inverses and a hadamard product?
I have a matrix equation of the form:
$$
A^{-1} = B + A \circ C
$$
where $\circ$ denotes the Hadamard product (i.e., $(A\circ C)_{ij} = A_{ij}B_{ij}$). How can I determine if a solution for $A$ ...
- 395
3
votes
1
answer
2k
views
Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices
I have an engineering back ground. Due to work, I came across this problem
\begin{align}
&\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\
s.t.~&\left(\mathbf{A}_0+\sum_{i=1}^{K}y_i\mathbf{A}_i\...
- 1,421
3
votes
1
answer
959
views
Condition number of a random 0-1 matrix
Consider a 0-1 integer $n \times n$ matrix with coefficients chosen uniformly over $\{0,1\}$. The probability that it is singular is exponentially small, and so we expect that it has a well-defined ...
- 1,444
4
votes
1
answer
860
views
Lebesgue's integrability condition in several variables
The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
- 21.8k
3
votes
1
answer
445
views
Is there an algorithm to compute group presentations of or find generators for the centralizer of a matrix in $GL(n, \mathbb{Z})$?
Let $M \in H \leq GL(n, \mathbb{Z})$. Is there an algorithm that computes either matrix generators or even a group presentation for $C_H(M)$ given generators or a presentation of $H$? Also is $C_H(M)...
- 31
2
votes
1
answer
474
views
When is there a solution to these coupled eigenvalue equations?
I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...
- 403
7
votes
1
answer
903
views
Does this cross-product norm inequality hold?
I asked this on MSE over a month ago, but the one answer I got doesn't seem to work.
Let $\times$ denote the cross-product. $\;$ Is it the case that
For all unit vectors $\:\mathbf{x}\hspace{.01 ...
user5810
21
votes
4
answers
9k
views
Condition for two matrices to share at least one eigenvector?
Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so $...
- 403
9
votes
2
answers
632
views
Conjugacy classes of PGL(3,Z)
We know that every $2\times 2$ matrix in $PGL(2, \mathbb{Z})$ of order $3$ is conjugate to the matrix $$ \left( \begin{array}{cc} 1 & -1 \\ 1 & 0 \end{array} \right) $$.
I am interested in ...
- 529
2
votes
3
answers
930
views
eigenvalue of Laplacian matrix
If we have a Laplacian matrix $\boldsymbol{A}$ such that
\begin{align}
&A_{ii} >0 \\
&A_{ii}=-\sum_{j\neq i}A_{ij}
\end{align}
with known eigenvalues $\lambda_i$.
Define the matrix $\...
- 29
1
vote
1
answer
172
views
Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]
Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds:
$\begin{pmatrix} a_n & & \\
\vdots & \ddots & \\
a_1 &...
- 398
7
votes
2
answers
605
views
Sum of Difference of anti-diagonal matrix elements
Let $A \in \mathbb{R}^{n \times n}$, with elements $a_{ij}$
What conditions on $A$ are required for the following to be true?
There exists some vector $x \in \mathbb{R}^n_+$, $x \neq 0$ such that ...
- 73
34
votes
6
answers
3k
views
Does seeing beyond the course you teach matter? The case of linear algebra and matrices
This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...
3
votes
1
answer
138
views
On matrices conjugated in a faithful representation
Let $k$ an algebraically closed field.
Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.
Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...
- 3,472
2
votes
0
answers
184
views
The minor of a square matrix
Given a $n \times n$ matrix, the $(i, j)$ minor is the determinant of the submatrix formed by deleting the i-th row and j-th column. If the sum of all row vectors and the sum of all column vectors are ...
- 517
3
votes
0
answers
562
views
Invariant subspaces of permutation matrix [closed]
Let $\sigma$ be a permutation matrix of order $n$. What are all the invariant subspaces of $\sigma$?
(I can only find 1 and n-1 dimensional subspaces)
Thanks in advance.
- 31
1
vote
1
answer
1k
views
Bounding the positive semi-definite matrix with its block diagonal matrix [closed]
Can we bound $\mathbf{A}$ with $\mathbf{A^*}$ as ${\bf{A}} \preceq {{\bf{A}}^*}$ where
\begin{equation}
{\bf{A}} = \left[ {\begin{array}{*{20}{c}}
{{{\bf{A}}_{11}}}&{...}&{{{\bf{A}}_{1N}}}\\
...
- 383
5
votes
0
answers
231
views
Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity
Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for $k=0,\dots,l-1$....
- 767
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
2
votes
0
answers
648
views
Closed-form expressions for dual norms of real normed vector spaces
Didn't get any biters over at MSE, so I figure this place might be more appropriate...
Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\...
- 4,907
4
votes
0
answers
836
views
Hermitian forms over quaternion algebra
Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let $\sigma=Int(i)\...
- 41