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

learn more… | top users | synonyms

2
votes
0answers
130 views

Separating the eigenvalues of a Hermitian matrix with a special block structure

I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where, $J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$ $A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$. $B \in ...
3
votes
0answers
68 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 ...
0
votes
0answers
43 views

Eigenvalues of a “Half-Kronecker ” Product

The Problem: Given a 2 by 2 matrix $C$(the matrix elements of C are given), and two other 2 by 2 matrices $A$ and $B$(the matrix elements of A and B are given). Now we can construct a new matrix ...
0
votes
0answers
45 views

Forming orthogonal bases in different orders

Let $\alpha_1, \dots, \alpha_n$ be unit vectors in some vector space $V = R^d$. For any permutation $\pi: [n] \rightarrow [n]$, we can form the Gram-Schmidt orthogonal bases $\beta_{\pi,1}, \dots, ...
2
votes
1answer
164 views

Matrix equation XAX=B where the solution must be diagonal [closed]

$$X_{solution}=\arg\min_X \|XAX-B|_F \quad\mathrm{subject\ to}$$ X is square and diagonal A is square and positive semi-definite B is square and positive semi-definite Any pointers or relevant ...
0
votes
0answers
45 views

Generalized Eigenvalue problem solution

$\circ$ Consider the following eigenvalue problem : $$XY^{T}YX^{T}w=\lambda XX^{T}w \hspace{0.5cm} (1)$$ Matrices $XY^{T}$ and $XX^{T}$ $\in \mathbb{R}_{n \times n}$ are positive semi-definite and ...
1
vote
0answers
163 views

An optimization problem on the sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer. Let vector ...
0
votes
0answers
81 views

Linear system with many solutions from a finite set

Basically I am looking for a linear system with many solutions from a finite set. Choose a finite set of rationals $S$ and fix positive integer $k$. Let $A$ be a linear system with $n$ variables ...
4
votes
2answers
157 views

Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows: $V_0(x) = 1-a^2$ $V_1(x) = 1-a^2 - x$ $V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$ ...
3
votes
1answer
97 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 ...
1
vote
2answers
82 views

Bound on smallest entry of inverse matrix

For a symmetric, invertible matrix $A=(a_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse ...
10
votes
1answer
206 views

Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if $$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$ But ...
3
votes
1answer
141 views

Calculating a generalized inverse (Moore–Penrose pseudoinverse)

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix: \begin{equation} A_n= \begin{pmatrix} 0 & 0 & 0 &\cdots & 0 & 0 & 1\\ 0 & 0 ...
1
vote
0answers
79 views

Dimension of $L(E,F)$

Let $E,F$ be two vector spaces over a field $k$. $L(E,F)$ is the $k$-vector space of linear maps $E \rightarrow F$. In ZFC, is there a functionnal relation between $dim(L(E,F))$ and ...
1
vote
2answers
95 views

Mixing Numerical Range and inner product

Let $\mathbf{A}$ and $\mathbf{b}$ be a symmetric $N\times N$ real matrix and $N\times 1$ real vector respectively. Then consider the set of points in $\mathbb{R}^2$ defined as \begin{align} ...
1
vote
0answers
91 views

Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I want ...
4
votes
0answers
75 views

Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
5
votes
1answer
91 views

Are there any known results on numerical ranges of rank-one positive semi-definite matrices?

In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then ...
4
votes
1answer
128 views

Finite generation of vector identities

This question is partially motivated by Looking for a comprehensive referece for vector identities, although that question may not be appropriate for MO. Consider the set $\mathcal{E}$ of all valid ...
6
votes
2answers
129 views

How to write this result (successive Schur complements compose nicely)

There is an easy theorem in linear algebra that "successive Schur complementations compose nicely": for instance, let's say I have a matrix $$ M_0= \begin{bmatrix} A & B & C \\ D & E & ...
3
votes
1answer
176 views

When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$. 1) For $t_k=k$, what is the condition on ...
0
votes
0answers
50 views

Linear Algebra: spectral decomposition by special single row replacement

Problems Statement: Let $\mathbf{Y}$ be an $m \times n$ real matrix whose $i$th row is $\mathbf{y}_i = \left(y_{i1},...,y_{in}\right)$ and whose rank is $n$. Let $\mathbf{y}_{i,\sigma }=\left( ...
1
vote
0answers
58 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
5
votes
4answers
260 views

Infinite matrix leading eigenvector problem

This question is cross-posted at Math.StackExchange.com. I'm trying to find the leading eigenvalue and corresponding left and right eigenvectors of the following infinite matrix, for $\lambda>0$: ...
14
votes
4answers
546 views

Smallest non-zero eigenvalue of a (0,1) matrix

What's the smallest absolute value possible of a non-zero eigenvalue of an $n$ by $n$ square matrix whose entries are either $0$ or $1$ (all operations are over $\mathbb{R}$)? I would be interested ...
0
votes
0answers
48 views

Cycles of Permutation Related to Rectangular Matrix Transposition

let the entries of a rectangular matrix $A\in\mathbb{C}^{m\times n}; m,n\in\mathbb{N}$ be stored in row-order in a linear vector $v$, i.e. $A_{i,j}=v_{i*m+j}$ Question: How can the first element ...
2
votes
1answer
19 views

Separations between notions of rank for modules over commutative (semi-)rings with no zero divisors.

Let $M$ be an $m$-by-$n$ matrix, here are three definitions$^5$ that we could use for rank: $rk(M) = \min k$ such that for matrices $P$, and $Q$ with $P$ of size $m$-by-$k$ and $Q$ of size ...
3
votes
1answer
78 views

Circuits in a linear oriented matroid

Given $E$ a finite subset of a real vector space $V$, a circuit of the associated matroid is a minimal linearly dependent subset of $E$. For each circuit $\underline C$, a minimal linear dependence ...
6
votes
1answer
238 views

Injectivity of matrix “fingerprint”

Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries. For any matrix $A$, define $$ ...
0
votes
0answers
47 views

A lower bound on the number of matrices whose image contains all multiples of $p^e$

Let $0\leq e<e^\prime$ be integers. Now suppose $N$ is the number of $n\times n$ matrices over the ring $R:=\mathbb{Z}/p^{e^\prime}\mathbb{Z}$ (where $p$ is prime) such that ...
3
votes
0answers
61 views

Orthonormal basis with small $\ell_{\infty}$ norm for a subspace

(Note: I posted this question on stackexchange but was told it might be more suited for mathoverflow. There has been some discussion about the problem there.) Let $U \subset \mathbb{R}^n$ be a ...
1
vote
0answers
112 views

Closed-form solution to a system of linear equations

Consider the following $n \times n$ matrix with a particularly nice structure: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 &1\\ 0 & 0& \dots&0 & ...
6
votes
1answer
159 views

Almost Hadamard matrices

As well-known, a Hadamard matrix is a square matrix with all coefficients $\pm 1$ and pairwise orthogonal rows or columns. Such matrices exist conjecturally in every dimension divisible by $4$. Call ...
0
votes
1answer
117 views

Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...
10
votes
0answers
215 views

Who stated and proved the “Hopf lemma” on bilinear maps?

If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$. Nondegenerate here means ...
2
votes
0answers
53 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 ...
3
votes
1answer
135 views

Reconstructing a (unitary) matrix from the determinant of its sub-matrices

I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ ...
0
votes
1answer
127 views

Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system. Let us we have an affine family of $n\times n$ matrices with indeterminate ...
1
vote
0answers
145 views

Incoherence of the row/column span

Due to V.Chandrasekaran., et al‎ (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that: $$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$ where the lower bound is achieved (for ...
2
votes
0answers
37 views

Convergence of Symmetric Iterative Proportional Fitting

Let $A$ denote a symmetric matrix of non-negative entries, whose rows (and columns) sum up to the all-positive vector $d := A\mathbf{1}$ with $\mathbf{1}$ the all-one-vector. Denote $D := diag(d)$ by ...
7
votes
2answers
800 views

Linear independence of the square roots over Q

Does there exist a real number $a$ such that the numbers $\sqrt{n^2 + a^2}$ (for all natural $n$) are linearly independent over the field of rational numbers? It is evident that $a$ cannot be ...
3
votes
0answers
105 views

Kasteleyn, Gessel-Viennot and eigenvalues

The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...
6
votes
0answers
295 views

On the number of polynomials that divide $x^{q-1}-1$ in some subspaces of $\mathbb F_q[x]$

Let $\mathbb F_q$ be a finite field (where $q$ is in general a power of a prime), and let $e, k$ be positive integers with $k \leq e < q-1$. Let $f_0(x), \ldots, f_k(x) \in \mathbb F_q[x]$ be ...
4
votes
1answer
218 views

Is there a generalization of linear algebra that allows fractional ranks?

The rank is the number of linearly independent rows/cols of a matrix. Generally, we think of linear independence as a binary property. But we could imagine an alternative definition that allows for ...
1
vote
2answers
180 views

Möbius transformation by 3 points in the Minkowski model

Goal I'm interested in describing Möbius transformations in the plane, and I'd like to define them in terms of three points and their images. What I have tried I know that a projective ...
11
votes
4answers
489 views

Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 ...
-1
votes
1answer
96 views

Inequality between two matrices

Given a full rank $n \times m$ matrix $K$ with $m<n$ and an invertible symmetric matrix $J$. Let $A$ be a symmetric positive semi-definite $n \times n$ matrix such that \begin{equation} (K^T ...
0
votes
1answer
52 views

Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$, $g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$, $g$ is ...
1
vote
1answer
107 views

Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field), $g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...
4
votes
1answer
127 views

Eigenvalues of generalized Vandermonde matrices

Given a strictly increasing sequence $0<x_1<x_2<\dots<x_n$ of $n$ strictly positive real numbers and a second strictly increasing sequence $e_1<\dots e_n$ of $n$ real numbers, the ...