Questions tagged [linear-algebra]

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

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Classification of algebras of finite global dimension via determinants of certain 0-1-matrices

I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background). A Morita-...
Mare's user avatar
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2 votes
2 answers
257 views

Equal-valued determinants in search of a proof: Part III

Encouraged by David's proof for my earlier MO question, let's consider a similar problem. I can prove the below equality by computing each of the two sides, directly. That means, there is an ...
T. Amdeberhan's user avatar
11 votes
1 answer
569 views

Catalan determinants in search of a proof: Part II

This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof. ...
T. Amdeberhan's user avatar
7 votes
1 answer
688 views

Is Gram-Schmidt on a separable Hilbert space operator norm continuous?

Let $\mathcal H$ be a separable Hilbert space, with inner product $\langle\cdot,\cdot\rangle$, and with orthonormal basis $(e_i)_{i\in\mathbb N}$. Consider a continuous linear embedding $A\colon\...
Benedikt Hunger's user avatar
5 votes
1 answer
407 views

Determining the primitive order of a binary matrix

Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows $$ {\bf A}_n=\left( \begin{array}{c} 0&0&\cdots&0&0&0&0&1&1\\ 0&0&\cdots&0&0&...
Amin235's user avatar
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5 votes
1 answer
1k views

eigenvalues of a symmetric matrix

I have a special $N\times N$ matrix with the following form. It is symmetric and zero row (and column) sums. $$K=\begin{bmatrix} k_{11} & -1 & \frac{-1}{2} & \frac{-1}{3} & \frac{-1}{4}...
A. doroudi's user avatar
7 votes
1 answer
496 views

Cycle types of permutations from affine group

Let $V$ be a vector space of dimension $n$ over the field $F=\mathrm{GF}(2)$. We identify $V$ with the set of columns of length $n$ over $F$. Let $G = \mathrm{AGL}(V)$ be a group of affine ...
Mikhail Goltvanitsa's user avatar
3 votes
1 answer
76 views

Dimension of fixed vectors of a semi-linear operator

Let $L$ be a field with a field embedding $\sigma:L \rightarrow L$, and $K=L^{\sigma}$ be the fixed field of $\sigma$. For $A \in M_n(L)$ a matrix, consider the set $X=\{x \in L^n|Ax=\sigma(x) \}$ ...
sawdada's user avatar
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4 votes
0 answers
380 views

Trace of the adjoint action is an eigenvalue in $\mathrm{U}(L)$?

Let $L$ be a finite-dimentional complex Lie algebra. $\forall x \in L$, one defines the adjoint action of $x$ on $L$ as the map $\mathrm{ad}_x : L \to L, \text{ with } \mathrm{ad}_x(y) = [x,y]$ for ...
rpz's user avatar
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8 votes
1 answer
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What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?

Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
Joe Fitzsimons's user avatar
0 votes
1 answer
242 views

Perturbing a normal matrix

Let $N$ be a normal matrix. Now I consider a perturbation of the matrix by another matrix $A.$ The perturbed matrix shall be called $M=N+A.$ Now assume there is a normalized vector $u$ such that $\...
user avatar
5 votes
1 answer
218 views

Stable matrices and their spectra

I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices. A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...
user avatar
2 votes
1 answer
186 views

Question on the proof of any finite dimensional module of a semisimple Lie algebra is semisimple

I'm going through a proof of the theorem, any finite dimensional module of a semisimple Lie algebra is semisimple, from page 10 of these notes (pdf). I am having a difficult time understanding few ...
Johnny T.'s user avatar
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6 votes
1 answer
416 views

If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis?

Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a ...
darij grinberg's user avatar
4 votes
1 answer
280 views

Which groups can be reconstructed from a single invariant subspace?

Let $G\subseteq\mathrm{Perm}(\Bbb R^n)$ be a matrix group consisting of permutation matrices acting on $\Bbb R^n$. Let $U\subseteq\Bbb R^n$ be an irreducible invariant subspace w.r.t. $G$. Now, define ...
M. Winter's user avatar
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1 vote
0 answers
54 views

When does the multi-spectral radius coincide with the spectral radius of the sum of linear transformations?

Suppose that $X$ is a finite dimensional Hilbert space and $A_{1},\dots,A_{r}:X\rightarrow X$ are linear transformations. Define the multi-spectral radius $\rho(A_{1},\dots,A_{r})$ to be $$\limsup_{n\...
Joseph Van Name's user avatar
2 votes
1 answer
485 views

An inequality on elementary symmetric polynomial of eigenvalues

For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds, $$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||_*^2 - ||A||_F^2$$ This is equivalent to the following inequality on ...
user1952770's user avatar
3 votes
1 answer
764 views

Real part of eigenvalues and Laplacian

I am working on imaging and I am a bit puzzled by the behaviour of this matrix: $$A:=\left( \begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 &...
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3 votes
1 answer
421 views

Spectrum of this block matrix

Consider the following block matrix $$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$ where all submatrices are square and matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
Sascha's user avatar
  • 506
0 votes
1 answer
115 views

Infinite norm of two randomly picked points [closed]

Let X and Y be points in the 4000-dimensional unit cube, picked at random with uniform distribution, which means from I what I understand that all locations in the cube are equally likely. $X \in [0,1]...
dxdydz's user avatar
  • 139
0 votes
0 answers
162 views

Smallest eigenvalues of block Kronecker product

Let $D \in \mathbb{R}^{n \times n}$ defined as \begin{equation} D := \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ -1 & 1 & \ddots & \ddots & 0 \\ \vdots & \ddots &...
JKay's user avatar
  • 133
3 votes
0 answers
151 views

Reference request: invariants/tableaux functions for 4 lines in $P^3$

Does anybody have a reference for invariants of configurations of linear subspaces in the projective space? In particular I would be curious to see an explicit expression of the invariant functions ...
IMeasy's user avatar
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5 votes
1 answer
202 views

Spectral radius for multiple linear operators

Suppose that $X$ is a finite dimensional Hilbert space. Let $A_{1},\dots,A_{r}:X\rightarrow X$ be linear operators. Then define the multi-spectral radius of $(A_{1},\dots,A_{r})$ to be $$\limsup_{n\...
Joseph Van Name's user avatar
0 votes
0 answers
227 views

What matrix has only negative or zero real part for all the eigenvalues?

Say $X \in \mathbb{R}^{m\times m}$, Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part? What I conjecture The following $X$ has only negative ...
ArtificiallyIntelligent's user avatar
3 votes
2 answers
1k views

What do the eigenvectors of the $n$th roots of $I_n$ look like?

This was asked at math stackexchange a long time ago with no answers but some upvotes. Let $A^n=I,$ where $A$ is $n\times n,$ and assume that $A^k\neq I,$ for all $1\leq k<n.$ Since its ...
kodlu's user avatar
  • 10.1k
3 votes
0 answers
254 views

Equal principal minors of matrix plus rank-1 and inverse

Given an invertible real matrix $A$ and real column vectors $b$ and $c$. For which $A$, $b$ and $c$ are all corresponding principal minors of $B = A-bc^T$ and $A^{-1}$ equal? According to a ...
Jiro's user avatar
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0 votes
0 answers
35 views

What is the locus defined by those equations?

I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by $\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$. I know that if $\...
MysteryGuy's user avatar
4 votes
2 answers
349 views

Linear operator on polynomials and invariant sets of roots

Let $T:\mathbb{C}_n[x] \to \mathbb{C}_n[x]$ be a linear map from the vector space of polynomials of degree $n$ to itself. Let $S \subset \mathbb{C}$ be a set with at least $3$ points, such that for ...
Per Alexandersson's user avatar
3 votes
1 answer
2k views

When is the matrix norm multiplicative

Let $|| = ||_{p,q}$ be an operator norm on $\mathbb R^{n \times m}$. In General, $\|AB\|\le \|A\|\|B\|$. Is there some criterion on $A, B$ (at least for some operator norms) so that $\|AB\| = \lVert ...
N_Segol's user avatar
  • 133
3 votes
1 answer
59 views

Looking for an Alternative Characterization of the Upper Orthant of a Convex Hull

I have a question on convex analysis, which I require in one step of my research work. Before stating it, let me give a small background. Let $Y, X_1,\ldots,X_n$ be $n+1$ points in $\mathbb{R}^d$. ...
Usermath's user avatar
3 votes
2 answers
1k views

Completely positive matrix with positive eigenvalue

A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$. All eigenvalues of $A$ ...
JKay's user avatar
  • 133
0 votes
1 answer
43 views

For an one-order Linear Recurrence of a vector sequence, does the corresponding item follow a Linear Recurrence? [closed]

Consider an one-order Linear Recurrence of a vector sequence, such as $${\bf x}_{n+1}={\bf A}{\bf x}_n$$ where ${\bf x}_n \in \mathbb{R}^m (\forall n)$, and ${\bf A} \in \mathbb{R}^{m\times m}$ and ${\...
zbh2047's user avatar
  • 601
10 votes
2 answers
979 views

Determinantal symmetry: proof requested: Part I

Consider the determinantal function $$F(a,b,c):=\det\left[\binom{i+j+a+b}{i+a}\right]_{i,j=0}^{c-1}.$$ I would like to ask: QUESTION. Can you provide an argument, combinatorial or otherwise, to ...
T. Amdeberhan's user avatar
5 votes
9 answers
7k views

Applications of basic linear algebra concepts to computer science? [closed]

I'm trying to explain linear algebra to some programmers with computer science backgrounds. They took a course on it long ago, but don't seem to remember much. They can follow basic formalism, but ...
Kim's user avatar
  • 4,034
8 votes
1 answer
472 views

A question about special linear group

Is there any way to find all matrices $G \in SL(n,\mathbb Z)$ such that there exists a matrix $A \in GL(n,\mathbb R)$ satisfying $$ AGA^{-1} \in SO(n,\mathbb R)? $$
Totoro's user avatar
  • 2,515
1 vote
0 answers
154 views

Properties of vector combinations in the non-negative orthant

Given a vector $x \in \mathbb{R}^{n}_{0+}$ such that $x = \sum^{k}_{i=1} \alpha_{i}v_{i}$, the vectors $(v_{1},...,v_{k}) \in \mathbb{R}^{n}_{0+}$ are an independent set, $k < n$, and $\alpha_{i} &...
nick.schachter's user avatar
2 votes
0 answers
183 views

Deriving category of quadratic functors

Recall that a fuctor $T: \cal C \to \cal A$ from pointed small $\cal C$ with coproducts to additive and Karoubian — or, even better, abelian $\cal A$ is called quadratic if kernel of sum of obvious ...
Denis T's user avatar
  • 4,416
8 votes
1 answer
170 views

Distance between subalgebras and positive elements in matrices

I repost here from stackexchange, as I was not given an answer there. (https://math.stackexchange.com/questions/2956530/distance-between-subalgebras-and-commutants-in-matrix-algebras) This is a ...
Alessandro Vignati's user avatar
6 votes
0 answers
367 views

Is there an easy way to compute the maximum isotropic subspace over finite fields?

Given a quadratic form (or a symmetric $n \times n$ matrix $A$), an isotropic subspace is a subspace $U$ such that $$U^t A U=0,$$ If I am not mistaken, when the matrix is over reals, the maximum ...
Hao's user avatar
  • 571
2 votes
0 answers
224 views

Quartic optimization problem over the unit Euclidean sphere

I want to solve following optimization problem in $x \in \mathbb R^n$. $$\begin{array}{ll} \text{maximize} & \displaystyle\sum_i(x M_i x^T)^2\\ \text{subject to} & \|x\|_2 = 1\end{array}$$ ...
makkostya's user avatar
  • 415
1 vote
1 answer
125 views

Expected norm of linear maps

I want to compute the expected norm of a vector-matrix multiplication. I have a vector $x \in \mathbb{R}^n$ with norm one and a matrix $M \in \mathbb{R}^{n \times n}$, whose entries are iid taken from ...
Alfred's user avatar
  • 879
0 votes
1 answer
142 views

Pre-garbling does not improve capacity of a channel

Suppose ABC=B for some column stochastic matrices A, B, and C. Can the following implication be made without further restrictions: There necessarily exists a column stochastic matrix D such that DB=BC?...
JayDoe's user avatar
  • 9
4 votes
2 answers
156 views

$\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting

Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$. Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality $$\left\|\displaystyle\...
Student's user avatar
  • 1,154
2 votes
1 answer
435 views

Coordinate free expression for the determinant of a $2 \times 2$ matrix

Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...
user avatar
-2 votes
1 answer
152 views

About intersections of two totally isotropic subspaces fo a quadratic form [closed]

Let $Q$ be a quadratic form on $\mathbb R^{2m}$ with the signature $(m,m)$. The maximal totally isotropic subspaces in $(\mathbb R^{2m},Q)$ have then dimensions $m$. What dimensions $1,...,m-1$ of ...
M.M2's user avatar
  • 11
2 votes
2 answers
111 views

Correlation between the first and a random position of an ergodic bit sequence

Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme. Probabilistic version. Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...
Ron P's user avatar
  • 947
5 votes
1 answer
93 views

On a maximum of a determinant with dependent variables

Let $x_1,\ldots,x_n\in [-1,1]^n$ and define the function $$f(x_1,\ldots,x_n):= \prod_{i=1}^n\prod_{j=i}^n\left(1-\prod_{k=i}^j x_k\right).$$ This is a positive function, and actually coincides with ...
F.Battistoni's user avatar
1 vote
1 answer
178 views

Subbundle generated by linearly dependent sections

On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is ...
user avatar
3 votes
1 answer
3k views

Why is the matrix of all 1's called "J"? [closed]

I've seen J referenced recently in some discussion about algebraic combinatorics, and it took me a while to figure out it was the matrix of all ones. It came up without definition, and I spent too ...
Joe Driscoll's user avatar
2 votes
0 answers
70 views

An two-norm estimate for symmetric $k$-tensors

Let $(V, \langle, \rangle)$ be an $n$ dimensional innerproduct space and let $S^k(V)$ denote the space of $k$-fold symmetric tensors. The inner product naturally extends to $S^k(V)$. Denote the ...
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