Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [linear-algebra]

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

2
votes
0answers
61 views

Show the spectral radius of a matrix is smaller than 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...
0
votes
0answers
33 views

Find the analytical form for the spectral radius of a special sparse matrix, or its order approaching 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...
1
vote
0answers
57 views

When are “square spans” not transversal?

Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...
2
votes
0answers
33 views

Volume interpretation of number of perfect matchings in bipartite planar graphs

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...
0
votes
1answer
103 views

Smallest collection of linear operators satisfying isometry property

Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...
6
votes
0answers
159 views

Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1)

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller ...
2
votes
0answers
69 views

How to formally connect the log-Euclidean and Riemannian metrics for Symmetric Positive Definite matrices?

I'm a statistician working on a research project dealing with metrics on SPD matrices, specifically the log-Euclidean $d_{LE}(X, Y) = \|\log(X) - \log(Y)\|$ and the Riemannian metric $d_{R}(X, Y) = \|\...
1
vote
1answer
78 views

A question on a special “metric”

Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...
3
votes
1answer
77 views

Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...
4
votes
1answer
332 views

Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums

The following question arose in some discussions recently as a misunderstanding of another problem. Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...
2
votes
1answer
122 views

Matrices with distinct columns

Let $M$ be an $n \times m$ matrix over $\mathbb{F}_2$ with no repeated columns, and suppose that $m \leq 2^{n-1}$: i.e., it is possible to have a matrix with fewer rows that still has $m$ unique ...
5
votes
1answer
230 views

Lefschetz operator

Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2\mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz: For $k\leq n$, the Lefschetz operator $L^{n-k}:\...
1
vote
1answer
103 views

Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic

This is a more elegant version of the original version which can be found below; it is based on a suggestion by Peter Mueller. Let $\mathbb{N}$ denote the set of positive integers and for $n\in\...
2
votes
1answer
179 views

$2$-adic valuation on $\mathbb Q (\sqrt{-15})$

I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that $-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ denotes ...
6
votes
1answer
204 views

Covering the finite plane with lines

This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form. Let ...
1
vote
1answer
230 views

How to verify the characteristic polynomial? [closed]

I am computing the characteristic polynomial of a matrix over a number field, using the minimal polynomial of it. Is there a fast way to verify the characteristic polynomial of a big matrix ?
4
votes
0answers
176 views

Forcing scalar products to avoid prescribed values

Let $p$ be a prime, and $n\ge 1$ an integer number. Suppose that the (not necessarily distinct) vectors $v_1,\dotsc,v_N \in{\mathbb F}_p^n$ satisfy the following condition: \begin{gather} \text{For ...
1
vote
0answers
74 views

Eigenvalues of non-negative block matrices

$B$ is a non-negative irreducible block matrix as follows: $B= \left[ \begin{array}{c|c|c} 0 &B_{12}&B_{13}\\ \hline B_{21}& 0& B_{23}\\ \hline B_{31}& B_{32}&0 \end{array} \...
2
votes
0answers
112 views

Maximal number of $S_n$-conjugates living in a hyperplane

Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, ...
1
vote
0answers
78 views

Generic deformation of matrix

Let $A(x)$ be a $m \times n$ matrix, whose entries are real polynomials $f_{i,j}:\mathbb{R}^S \to \mathbb{R}$. Denote the ith row by $f_i$ And let $rk:\mathbb{R}^S \to \mathbb{N}$ be the rank function ...
2
votes
1answer
145 views

Linear subspaces in quadric hypersurfaces

Consider $H_1,H_2,H_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$. Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,...
4
votes
1answer
75 views

Separate the trivial partition by a linear hyperspace

Let $e=[1,1,\ldots,1]\in\mathbb{Z}^n$. I am looking for a way to find a vector $a\in\mathbb{Z}^n$ such that: $\langle a,e\rangle=0$ and for every nonnegative $v\in\mathbb{Z}^n$ such that $\langle e,v\...
2
votes
0answers
44 views

Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

This is cross post to the question at MSE. Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also ...
11
votes
1answer
314 views

Decide if a matrix is transposable

A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations. Is there an efficient a way/algorithm to decide if a given matrix is ...
2
votes
0answers
26 views

Approximate Simultaneous Diagonalization of Non-Hermitian Matrices

Let $A_1,A_2$ two $n\times n$ complex matrices. $A_1$ and $A_2$ are also non-normal, especially, non-hermitian and do not commute. I would like to find an invertible matrix $V$ such that $$ \sum_{i=1,...
8
votes
3answers
205 views

Determinant of a block matrix with many $-1$'s

For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows: The main diagonal has blocks of sizes $n_i$ and ...
1
vote
0answers
33 views

Choice of residual function for least squares error minimization

Good morning, I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data. I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as $K_{IC} = \sigma \sqrt{D} k_0(\...
20
votes
2answers
445 views

$GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class

It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...
4
votes
2answers
171 views

The maximal size of intersection of two sets

Let $S=\{1,2,\cdots,2n\}$, and $S_i \subseteq S(i=1,2,\cdots,n+1)$ be $n+1$ subsets, each of which contains half of the $2n$ elements, namely $|S_i|=n$. Consider the following expression: $$M=\max_{1\...
3
votes
0answers
194 views

Matrix Inequality: Traces of $n$th powers

Let $A, B$ be matrices over $\mathbb{C}$ of the same dimensions (not necessarily square). With $'$ denoting conjugate-transpose, and tr the trace, show for $n\in\mathbb{N}$ that $ 2\,\mathrm{Re}\, \...
0
votes
1answer
97 views

About roots of the equation $det(A-\lambda B)=0$ [closed]

Let $A,B$ be square real symmetric matrics of the same degree $n \geq 3$ without common isotropics vectors (i.e. there is no a nonzero vector $x\in \mathbb R^n$ such that $x^TAx=x^TBx=0$). Are roots ...
1
vote
0answers
77 views

Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?

Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no. At least could it be true in $2\times2$ ...
6
votes
1answer
141 views

Bounding the eigenvalues of $B A B^T$ with the eigenvalues of $A$

Given a Hermitian positive semi-definite $n \times n$ matrix $A$ and a rectangular $m \times n$ matrix $B$, is there anything that can be said about the eigenvalues of the matrix $B A B^T$? It seems ...
2
votes
0answers
65 views

What is the most efficient path for a robot without turning radius?

I recently programmed a most efficient path for a robot going from $(x_1, y_1 , \theta_1)$ to $(x_2, y_2)$. The bot does a combination of turning and moving at any given point, based on the difference ...
0
votes
1answer
256 views

Conjecture that relates matrix systems with some specific functions as solution sets

what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...
8
votes
1answer
181 views

Making binary matrix positive semidefinite by switching signs

Let $A \in \{\pm 1\}^{n \times n}$ be a symmetric matrix whose diagonal entries are $+1$. Let $f(A)$ be the smallest number of signs we need to change in $A$ so that it becomes positive semidefinite (...
6
votes
2answers
181 views

Nonlinear boolean functions

Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
6
votes
2answers
171 views

Degree of the variety of independent matrices of rank $\leq r$?

Consider an $m$-by-$n$ matrix $A$ with entries in a field $k$; we can see $A$ as a point in the affine space $\mathbb{A}^{m n}$. The rank of $A$ will be $\leq r$ (where $r<\min(m,n)$) if and only ...
0
votes
0answers
53 views

Can a rational symmetric matrix $A$ be diagonalized as $A = P \Lambda S$ for some $P, S $ in the general linear group?

Let $A$ be a $3 \times 3$ symmetric matrix with rational entries. Does there exists a unique pair of matrices $P, S \in \text{GL}_3(\mathbb{Z} )$ depending on $A$ such that $A = P \Lambda S$, where $\...
1
vote
0answers
30 views

Defining a notion of “volume of its lattice” for non-rational subspaces

Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”: $$\...
1
vote
2answers
154 views

How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?

$C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists? I am interested in this question, because my ...
2
votes
1answer
144 views

The effect of random projections on matrices

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$. Suppose $\frac{A+...
4
votes
2answers
453 views

Reference request: Oldest linear algebra books with exercises?

Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
7
votes
2answers
248 views

How do fractional tensor products work?

[I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...
2
votes
1answer
232 views

A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define $A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
1
vote
1answer
49 views

Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

This is a cross-post to the question I asked at MSE. The set of Schur stable matrices is \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where $\rho(\cdot)$ denotes ...
2
votes
1answer
141 views

Irreducible but not completely irreducible

Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <\infty$). Let $F:M\times \mathbb R^d\to M\times \mathbb R^d$ be a linear cocycle, ...
3
votes
2answers
102 views

Eigenvalue density of a symmetric tridiagonal matrix

Let $A_n\in\mathbb{R}^{n\times n}$ be defined as $$ A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
1
vote
1answer
81 views

Can eigenvectors determine their original matrices given the basis of matrices space is small?

Let $A_1$, $A_2$, and $A_3$ be three different mutually orthogonal random hermitian operators, say dimension of $30\times 30$. For three random real numbers $a_1$, $a_2$, $a_3$, we have $$ (a_1A_1+...
1
vote
1answer
37 views

Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system: \begin{equation} \mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1] \end{equation} I would adapt the definition of marginal stability from this ...