Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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7
votes
1answer
767 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 $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
5
votes
3answers
540 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
1answer
79 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 ...
4
votes
2answers
228 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 = ...
2
votes
0answers
100 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, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...
0
votes
0answers
207 views

Matrix inversion lemma and pseudoinverse?

Starting with this form of the matrix inversion lemma $$ (A^T B^{-1} A + D^{-1})^{-1} A^T B^{-1} = D A^T (B + A D A^T)^{-1} $$ substitute $B \rightarrow I$, $D \rightarrow \alpha^{-1} I$, $$ ...
4
votes
3answers
202 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 ...
0
votes
0answers
192 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 ...
5
votes
3answers
290 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 ...
2
votes
1answer
82 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 ...
2
votes
0answers
105 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 ...
0
votes
1answer
64 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
0answers
187 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 ...
0
votes
0answers
60 views

Constructing a digraph from its spectrum

This is related to the following question from cs theory stack exchange: http://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem So it seems as if given a sequence of real ...
5
votes
1answer
202 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 ...
2
votes
0answers
134 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 ...
3
votes
1answer
135 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 ...
2
votes
0answers
120 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, ...
6
votes
2answers
218 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 ...
11
votes
2answers
401 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 ...
2
votes
3answers
530 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 ...
4
votes
1answer
217 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 ...
5
votes
1answer
304 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 ...
4
votes
0answers
68 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 ...
4
votes
1answer
105 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 ...
2
votes
2answers
112 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 ...
8
votes
3answers
1k 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 ...
6
votes
4answers
462 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 ...
2
votes
1answer
155 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 ...
4
votes
0answers
100 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 $\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...
2
votes
1answer
110 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 $A$ ...
3
votes
1answer
156 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 \\\ ...
3
votes
1answer
119 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 ...
3
votes
1answer
117 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 ...
2
votes
1answer
138 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 ...
5
votes
1answer
224 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 ...
14
votes
4answers
792 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 ...
5
votes
2answers
356 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 ...
2
votes
3answers
231 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 ...
2
votes
1answer
105 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 ...
7
votes
2answers
165 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 ...
23
votes
6answers
1k 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
1answer
118 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 ...
0
votes
0answers
163 views

Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims: Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...
2
votes
0answers
68 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 ...
0
votes
0answers
88 views

Operator Adjoints and Non-Symmetric Inner Products

Let $V$ be a finite dimensional vector space (over $C$ if that makes a difference), and let $T$ be a linear operator on $V$. Now if $(\cdot,\cdot)$ is an inner product on $V$, then it is well-known ...
2
votes
0answers
128 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.
0
votes
1answer
141 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}}}\\ ...
5
votes
0answers
191 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 ...
3
votes
2answers
203 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 ...