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Questions tagged [linear-algebra]

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

13
votes
0answers
289 views

Cardinality vs. isomorphism type of vector spaces without choice

One of the classical uses of the existence of bases of vector spaces (which is equivalent to the axiom of choice) is the following theorem: If $V$ is an infinite vector space over a field $F$, and $...
2
votes
0answers
170 views

A parametrization of stable matrices

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$. My question. I'm wondering whether it is ...
-1
votes
1answer
70 views

How many hyper-rectangle-like objects are intersecting a hyperplane?

Let $A\in \mathbb R^{n\times n},\ b\in \mathbb R^n$ such that $\forall x\in \{-1,1\}^n : Ax\ne b$. Let us denote: $S=\{x\in\mathbb R^n|Ax=b\}$ ('S' for solution set). Is $\ \#\Big\{H\in\big\{ \{-1\},...
4
votes
1answer
163 views

higher Casimirs for $\mathfrak{sl}$

The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
4
votes
1answer
160 views

Which inner products preserve positive correlation?

Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there ...
7
votes
1answer
273 views

Block matrices and their determinants

For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows: (a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...
2
votes
0answers
146 views

Evaluate a curious determinant with Legendre symbol entries

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. R. Chapman's conjecture on the exact value of the determinant of $$C_p:=\left[\left(\frac{i-j}p\right)\right]_{0\le i,j\le (p-...
7
votes
1answer
396 views

A new determinant question for primes $p\equiv3\pmod4$

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by my question http://mathoverflow.net/questions/310301, here I introduce the matrices $A^+_p$ and $A^-_p$ ...
3
votes
1answer
138 views

Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...
13
votes
0answers
277 views

A determinant problem for primes $p\equiv 1\pmod4$

Let $p$ be an odd prime, and let $A_p$ denote the matrix $$[a_{ij}]_{1\le i,j\le (p-1)/2},$$ where $$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...
2
votes
0answers
85 views

An inequality concerning the solution of a Lyapunov equation

Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
13
votes
3answers
777 views

Is $-\det\big[\big(\frac{i^2+j^2}p\big)\big]_{1\le i,j\le (p-1)/2}$ always a square for each prime $p\equiv 3\pmod 4$?

Let $p$ be an odd prime and let $S_p$ denote the determinant $$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$ with $(\frac{\cdot}p)$ the Legendre symbol. By Theorem 1.2 of my ...
5
votes
0answers
160 views

Determinant arising in a problem from probability

Consider the determinant: $$\Delta:= \left|\begin{array}{cccc} A_{j_1} & A_{k_1} & A_{j_1}A_{k_1} & 1 \\ A_{j_2} & A_{k_2} & A_{j_2}A_{k_2} & 1 \\ A_{j_3 } & A_{k_3 } &...
4
votes
2answers
246 views

Fast projection onto a subspace

Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...
1
vote
2answers
104 views

fast way to calculate normal to set of vectors with $\pm$1 entries

Say I have a set of $(n-1)$ linearly independent vectors $\mathbf{v}_i$ of dimension $n$ with entries $\pm1$. I am interested in finding the $n-$dimensional vector $\mathbf{u} $which is normal to the ...
6
votes
1answer
221 views

Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?

Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent: The all-one vector $j$ is contained in the conic hull of $col(A)$. ...
2
votes
1answer
76 views

What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?

I have two problems related to eigenvalues of negative definite matrices: I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M_{11}-M_{12}M_{22}^{-1}M_{21}$ by taking $M=[...
1
vote
0answers
62 views

Determinant and restriction of scalar

Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its ...
2
votes
0answers
54 views

Can projecting a simplex onto orthogonal subspaces exposes the same vertices and edges?

Given the regular $n$-dimensional simplex $S\subset\Bbb R^n$ with $n\ge 4$, as well as two orthogonal subspaces $V,W\subset\Bbb R^n$ of dimension $\ge2$ (not necessarily of same dimension, not ...
9
votes
2answers
392 views

A trace-constrained maximization problem in the cone of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
5
votes
1answer
233 views

Largest Eigenvalue of a Matrix with Special Form in terms of n

In one step of solving a difficult problem, I would like to know the largest eigenvalue of a matrix with this pattern: $$A_n = \begin{bmatrix} 0 & 0 & 0 & 0 &\dots & 0 \\ ...
4
votes
3answers
462 views

Non linear matrix equation

I want to solve the following non linear matrix equation for $X\in\mathbb{R}^{N\times N}$: \begin{equation} XX^{\top}+ABX^{\top}-A=0 \qquad (1) \end{equation} For a given matrices $A\in\mathbb{R}^{...
4
votes
2answers
144 views

Hyperrectangle that contains most of cube's interior (except its vertices)

Let $n>0$, and let $p,q\in (0,1)$ such that $p<q$. Is there a hyperrectangle $H$ that satisfies the following: $\forall i\in{1,\dots,n}:\\ H\supset \prod_{j=1,\dots,n} \begin{cases} [p,q], &...
11
votes
0answers
191 views

Matrices that admit a power that is symmetric

We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
3
votes
2answers
122 views

Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$

I am working in data science and I have to deal with the following problem for which I would like to find a simplification: We call a function almost positive if $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,...
0
votes
1answer
79 views

Name of a matrix with one column and row removed [closed]

I am looking for the exact name of a matrix where the i-th column and rows have been removed. I cannot remember how it is called in linear algebra, does anyone got an idea? Thanks!
3
votes
0answers
105 views

An eigenvalue of certain family of matrices

Consider the matrices $$M_n=\left[\binom{i}j+\binom{2n+1-i}{j-i}+\binom{2n+1-i}j\right]_{i,j=0}^n.$$ I am convinced and hence would like to ask: Question: Is $0$ an eigenvalue of $M_n$?
3
votes
1answer
107 views

Reference on completely positive maps which are isometries

Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
6
votes
1answer
118 views

Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?

We know iteration ${\mathbf X}_k=\mathbf{A}{\mathbf X}_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see here). Is there any known rule for iteration ${\mathbf X}_k={\...
2
votes
0answers
56 views

Recovering a rank-one matrix from its eigendecomposition after randomized rounding

Let $A = xy^T$ be a rank-$1$ matrix, and suppose every entry of $A$ is in $[0,1]$. We can create a binary matrix $A_{\rm rounded}$ by setting $$ [A_{\rm rounded}]_{ij} = \begin{cases} 1 & \mbox{ ...
3
votes
0answers
78 views

Simultaneous Congruence of Two Matrices

Could you please let me know an answer to the following question on simultaneous congruence of two matrices. This question came up while trying to handle a system of PDE. QUESTION: Let $A,B\in M_n(...
3
votes
0answers
49 views

Linearity of a canonical morphism related to scalar extension and coextension

Let $h\colon R\rightarrow S$ be a morphism of commutative ring. Let $M$ and $N$ be $R$-modules. We consider the canonical morphism of $R$-modules $$p\colon{\rm Hom}_R(S\otimes_RM,N)\rightarrow{\rm Hom}...
3
votes
2answers
184 views

Upper bound of spectral radius of the sum of two matrices, one with spectral radius no larger than 1, and the other has small eigenvalues

Suppose I have one $pN\times pN$ matrix $\bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $\bf H$ is in a block-like ...
2
votes
0answers
55 views

Number of distinct directions in the set $\mathbb{Z}^2 \cap B(0,R)$

We say that non-zero directions $v, w \in \mathbb{R}^2$ are equivalent if they span the same line (i.e. $\exists C \in \mathbb{R}: v = Cw$.), and distinct otherwise. Given a collection $V \subset \...
1
vote
0answers
100 views

Bounds of Procrustes problem

We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following: \begin{align} f(A,B)...
9
votes
0answers
160 views

Maximum dimension of a space of $n\times n$ real matrices with at least $k$ nonzero eigenvalues

Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every ...
1
vote
0answers
59 views

Gram determinant in dual basis [duplicate]

Assume that $V$ is a $n$ -dimensional inner product space. Basis $(e_1,...,e_n)$, $(f_1,...,f_n)$ are said to be dual if $\langle e_i, f_j\rangle=0$ for $i\neq j$, $i,j=1,...,n$ and $\langle e_i, ...
0
votes
1answer
80 views

Degree of roots of unity in the spectrum of an integer matrix

Let $A$ be an $n\times n$ integer matrix whose all eigenvalues are roots of unity. It is known that for $n=2$, the degrees of these roots can be $1,2,3,4$, or $6$. What are the degrees for arbitary $...
0
votes
0answers
83 views

Matrices with half columns nonnegative and half columns nonpositive real numbers

Suppose I want to study square even dimensional matrices that have the following property: Half of the columns have nonnegative entries and the other half have nonpositive entries. Now given an ...
1
vote
1answer
104 views

Walks of odd Lengths in a Matrix

Consider the following matrix $$ A=\left[ \begin {array}{cccc} 1&1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ 1&0&0&0 \end {array} \right]. $$ Assume that $B=A^k$ ...
4
votes
1answer
141 views

How to find the analytical representation of eigenvalues of the matrix $G$?

I have the following matrix arising when I tried to discretize the Green function, now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
0
votes
1answer
66 views

Computing spectrum of convex combination of SPD matrices given individual spectral decompositions

Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
17
votes
2answers
423 views

On a special type of normed linear spaces

Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying $$ \|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z} $$ is a group ...
1
vote
0answers
53 views

Linear dependence of solution?

Consider the function $f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$ In the following we want to study the ...
1
vote
1answer
117 views

Lanczos algorithm for finding $k$ smallest eigenvector

I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
4
votes
3answers
207 views

Extending a continuous map over projective space

Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
3
votes
1answer
158 views

If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [duplicate]

Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
-2
votes
1answer
100 views

What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]

Suppose we have the following symmetric matrix. $$A = \sigma^2 I + u u^T$$ What can we say about the eigendecomposition of $A$?
5
votes
2answers
135 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$. Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
18
votes
1answer
1k views

A linear algebra problem in positive characteristic

Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...