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

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-3
votes
0answers
13 views

Algebra II exercise (help) [on hold]

Can anyone please help me with this exercise for my exam? It says: 1)Given S={(x₁,x₂,x₃) ∈ R^3 : x₁ + x₂ -2x₃ = 0}. a)Prove that S is a subspace. b)For each of the matrices A shown, check if ...
0
votes
0answers
11 views

Positive Elements of a $\ast$-Algebra

In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...
1
vote
1answer
27 views

Set of density matrices

A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^*=A; \operatorname{tr}(A)=1; A \ge 0\}.$ In Quantum Mechanics it is natural to look at a group action $\Phi: ...
2
votes
0answers
39 views

Polynomials with positive coefficients passing through fixed points/range of Vandermonde matrices

I'll give two equivalent statements of the setup, then give my questions. Fix integers $M \leq N$ and define the Vandermonde-like matrix $V_{M,N}[i,j] = (1 - \frac{i}{M})^{j-1}$ for $i \in ...
-3
votes
0answers
37 views

eigenvalues of cycle and its complement [on hold]

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...
-4
votes
0answers
25 views

What is the upper bound for training Linear separable set with Perceptron, Rosenblatt rule? [on hold]

I have the following neural networks problem and couldnt find any answer on the web. Any hints would help. I am not looking for a complete solution, just some pointing in the right direction. ...
-1
votes
0answers
37 views

A question on spectrum [migrated]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
-3
votes
0answers
40 views

Decomposition of orthogonal matrix into 2 orthogonal matrices [on hold]

Is there anyway to find a decomposition of orthogonal matrix $A$ into 2 orthogonal matrices $P$ and $Q$ such that $A = PQ^T$?
-1
votes
0answers
56 views

Interpolating Product of two Polynomials

Consider we have two non-constant polynomials $A(x)$ and $B(x)$. We define the polynomials over field $\mathbb{Z}_p$, for a large prime number $p$. We define Polynomial $A(x)$ as follows: ...
0
votes
0answers
27 views

Interpolating a polynomial when we permute part of $y_i$'s

Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and they are ...
4
votes
0answers
40 views

$AXB$ sort of decomposition? [migrated]

Let $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1 \dots, B_d ...
4
votes
0answers
145 views

Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
1
vote
0answers
63 views

Interpolating a Polynomial Given Multiplier of each $y_i$

It would be great even if you answer only one of the below questions. We have polynomial $P(x)=(x-\beta)\cdot g(x)$, where degree of $P(x)$ is fixed n-1, $\beta$ chosen uniformly at random from the ...
0
votes
2answers
94 views

Estimating the shift in the $\lambda_{max}$ of a matrix under a diagonal perturbation

Given a matrix $A$ and a diagonal matrix $D$, what ways do we have to estimate, $\lambda_{max}(A+D) - \lambda_{max}(A)$? (Feel free to make other assumptions about the matrices that they are all ...
-5
votes
0answers
40 views

Function that outputs only 1 or 0 depending on sign of variable? [closed]

Is there a single variable (preferably simple) function which equals 0 for any positive input and 1 for any negative input, or vice versa?
1
vote
0answers
48 views

Avoiding the range of a bivariate integer function or Diophantine function [on hold]

I have a bivariate integer function $f(x,y)=5+23x+7y+30xy$ where $x,y \geq 0$ and are integers. The lattice points of this function, or its range contain a large number of values. I'm trying to see if ...
3
votes
1answer
101 views

Number of linearly bisected subsets in finite vector space $F_2^n$

We consider the $n$-dimenstional finite vector space $\mathbb{F}_2^{n}$ over the finite field of two elements. For a subset $A\subseteq \mathbb{F}_2^{n}$ of even size $|A|=2m$ and a linear form ...
3
votes
1answer
153 views

On the search for an explicit form of a particular integral

Let $f$ be integrable over the interval $(0, 1)$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
4
votes
0answers
86 views

An inequality from the “Interlacing-1” paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132) For the argument to ...
-3
votes
0answers
46 views

How to find $B$ by solving the following linear system: $s_k$ $B$ ${s_k}^T$ $=1,$ [closed]

How to find $B$ by solving the following linear system: $s_k$ $B$ ${s_k}^T$ $=1,$ $\qquad$ for $k=1 ... ,p$. Where $s_k$ is a $1\times3$ row_vector from the matrix $S= [s_1 ... ...
0
votes
0answers
66 views

A question in compact set

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
0
votes
0answers
46 views

Equivalence classes of pairs linear transformations

Consider the set of 4-tuples: $$S_{(x, y), k} = \{ (a_i, b_i, a_j, b_j) : \|a_ixb_i - a_jyb_j\|_F^2 = k \}$$ for $a \in GL(m, \mathbb{R})$, $b \in GL(n, \mathbb{R})$, $x, y \in \mathbb{R}^{m \times ...
5
votes
0answers
99 views

Biggest (or large) rectangle in a polytope

I need an efficient method to construct a (hyper)rectangle inside a polytope with a lot of dimensions (say $100 < d < 1000$). Ideally I'd want the biggest possible rectangle, but as I don't ...
15
votes
2answers
497 views

Matrix equation $XAXBXC=I$

Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution ...
6
votes
0answers
151 views

When is a polynomial ring free over a graded subalgebra?

Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...
-1
votes
0answers
47 views

Orbit closures of symmetric bilinear form [migrated]

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where: ...
4
votes
3answers
134 views

Is an associative division algebra required for this phenomenon?

For which integers $d \geq 1$ can we find real matrices $R_1, \dotsc, R_d$ of size $d \times d$ such that for any unit vector $v \in \mathbb{R}^d$, $$R_1 v, \dotsc, R_d v$$ is an orthonormal basis? ...
5
votes
0answers
121 views

Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$

Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is \begin{equation} \Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in ...
2
votes
1answer
256 views

Is there an algorithm to test whether a vector is an eigenvector of a power of a matrix?

Given a square matrix $A\in k^{n\times n}$ and a vector $x\in k^n$ over some field $k$, is there an algorithm to test whether there are $s\in\mathbb{N}$ and $\lambda\in k$ such that $A^sx=\lambda x$? ...
4
votes
2answers
69 views

Integral roots of circulant matrix

When does the circulant matrix have only integral roots? For example: all roots of the adjacency matrix of the complete graph $K_n$ are integer, which its adjacency matrix is circulant, but in case ...
0
votes
1answer
82 views

Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: ...
2
votes
0answers
185 views

Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way. All the ${\lambda}_i$ are distributed the same way with chi-square ...
0
votes
0answers
24 views

The Condition Number of a Scaled Vandermonde

Let $V(x_1,..,x_n)$ be the Vandermonde induced by $x_1,..,x_n$ and Let $\tilde{V} = V(\frac{x_1}{h},...,\frac{x_n}{h})$. My intuition says that the condition number should be invariant to such ...
3
votes
1answer
204 views

Does there exist a norm on continuous real-valued function space?

I know the space of continuous real-valued function on closed set can be given a norm by integral. How about the continuous funcion on the real line? It may be non-integrable, like f(x)=x^2. So, does ...
1
vote
0answers
74 views

Can we have extension of Mercer theorem to interpolation? [closed]

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...
0
votes
1answer
88 views

Reference request: Strong Connectivity and the Incidence Matrix

Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix? Details: Consider a digraph $(V, E)$ with vertex set $$V = ...
1
vote
0answers
98 views

Maximizing sum of matrices

Over the last few months, I've been trying to find the solution to a research-related problem I'm having. However, my research is not in mathematics, and my progress toward reaching a solution has ...
1
vote
1answer
87 views

Transversality in Morse theory, linear algebra version

I am working on a product in Morse-Bott homology which has led me to the following considerations and unanswered question. I would be very grateful if anyone could help. Suppose $H:\mathbb{R}^n \to ...
12
votes
1answer
407 views

Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$

Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$. EDIT: I forgot to add that no ...
1
vote
0answers
49 views

Finding optimal linear transformation for intersection of convex polytopes

I previously posted this on MathSE and am now trying here. I have the following situation, as shown in the following diagram: $W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in ...
3
votes
1answer
207 views

Polynomial with the smallest area

Let $P_n(t) = p_0 + p_1 t + \cdots + p_n t^n$ be a polynomial (with real coefficients) of degree $n$ in the variable $t$. I am interested in the quantity $$\Phi_n = \min_{\sum_{i=1}^n p_i^2 = 1} ...
6
votes
1answer
132 views

Varieties parametrizing skew-symmetric matrices

Let $V$ be a vector space of dimension $n$ and let us consider the projective space $\mathbb{P}(\bigwedge^2V)$ parametrizing skew-symmetric matrices. Let $M\in\mathbb{P}(\bigwedge^2V)$, for any ...
-1
votes
0answers
49 views

Spectral radius of principal submatrices for the case of hermitian matrix

A principal submatrix of a matrix $\mathbf{A}\in\mathbb{C}^{N\times N}$ is any submatrix from $\mathbf{A}$ for which the same rows and columns have been eliminated. Assume $\mathbf{A}$ is hermitian, ...
2
votes
0answers
100 views

Rank–nullity theorem for finite von Neumann algebras

The rank-nullity theorem states that for $U, V$ finite dimensional vector spaces and $T:U \to V$ a linear map $$\dim(U) = \dim(im(T)) + \dim(ker(T)) $$ Let $M \subset B(H) $ be a finite von Neumann ...
0
votes
1answer
75 views

The weird projection from SO(2n)/B to maximal isotropic grassmannian

Take the generalized flag variety $SO(2n,\mathbb{C})/B$, considered as the moduli of isotropic flags (according to the form $\langle e_i, e_{2n+1-j}\rangle=\delta_{ij}$) $$F_1\subset F_2\subset\cdots ...
5
votes
4answers
361 views

Deceptive linear algebra problem

Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before: Given $d_1, d_2, \ldots d_p$, find $a_1, a_2, \ldots ...
0
votes
1answer
58 views

Possible lower bound in quantum many body system with non-local terms

I am asking a question related to Lieb-Robinson bound and nonlocality. As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. ...
2
votes
1answer
100 views

Sampling from random totally unimodular matrices of a particular type?

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form $$\begin{bmatrix} \pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\ A_{2n} & ...
6
votes
2answers
112 views

Existence and characterization of transitive matrices?

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following: For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw ...
2
votes
0answers
99 views

Vector inequation problem [closed]

$${A_i} = \left( {\begin{array}{*{20}{c}}{{A_{i1}}}\\{{A_{i2}}}\\ \vdots \\{{A_{in}}}\end{array}} \right),{B_i} = \left( {\begin{array}{*{20}{c}}{{B_{i1}}}\\{{B_{i2}}}\\ \vdots ...