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

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3
votes
2answers
85 views

Complexity of solving systems of linear diophantine equations

It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...
2
votes
0answers
118 views

Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
10
votes
1answer
434 views

Pfaffian equals complex determinant?

Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure $$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}....
2
votes
0answers
72 views

What are the eigenvalues of a square matrix added to a diagonal matrix with different diagonal elements?

Suppose that we have a positive definite matrix $A$ of size $n$ and a diagonal matrix $D$ with only two different eigenvalues $d_1$ and $d_2$ on its diagonal. We know the eigenvalues and eigenvectors ...
2
votes
1answer
91 views

multiplicity-free action on $SO(n+1)/SO(n-1)$

I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^* (SO(n+1)/SO(n-1))$. That means: 1) There exists an $\operatorname{Ad}^*_G$-...
1
vote
2answers
96 views

Spectral radius of a non-negative matrix after moving and replicating an element

Let $A$ be a non-negative square matrix and its spectral radius (i.e., it's largest eigenvalue) be $\rho(A)$. I need to do the following operation to $A$ and compare the resulting spectral radii. ...
7
votes
2answers
240 views

How to estimate a specific infinite matrix sum

Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
1
vote
1answer
102 views

Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis. Suppose there is a matrix $$ A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] \...
1
vote
0answers
74 views

Number of positive eigenvalues of the product of two matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix that has at least one positive eigenvalue with positive real part. Let $B\in\mathbb{R}^{n\times n}$ be a matrix where all its eigenvalues have positive ...
4
votes
1answer
134 views

Existence of Nonnegative Solutions of Linear Systems of Equations and Inequalities with particular constraints

Suppose we have an $n \times m$ nonnegative matrix $A$, where each row sums to $1$. I wonder whether there exists an $m \times n$ nonnegative matrix $X$ that satisfies the following constraints: ...
3
votes
1answer
80 views

Transitivity of $Spin(7)$ in triples of vectors

I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e_1,e_2,e_3$, $v_1,v_2,v_3$ be two triples of mutually ...
-6
votes
1answer
57 views

Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?

Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true? $\|A\|_{2}$ denotes ...
3
votes
0answers
94 views

How do I ensure that my matrix is positive definite? [closed]

I require a positive definite matrix, $M$, with dimensions $2n\times2n$ of the form \begin{equation} M=\begin{pmatrix} \Sigma&P'\\ P&\Sigma \end{pmatrix} \end{equation} where $\Sigma$ is a $n\...
0
votes
0answers
89 views

Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers. Does $$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in \...
1
vote
1answer
84 views

Symmetric tensors as sum of powers

I am looking for formulas for writing a basis element of $ Sym^k(H) $ as sum of elements of the form $ v^{\otimes k} $ where $ v\in H $. Here $ H $ is a hilbert space and by basis element I mean the ...
2
votes
0answers
64 views

A problem of defining addition in a Quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Let the set of all re-parameterizations of curves is $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (...
1
vote
1answer
92 views

Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists. Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory). Preferably the ...
1
vote
1answer
44 views

Dual lattices up to a q scaling factor

In this paper : https://eprint.iacr.org/2011/501.pdf There is an equality page 10, in the second paragraph considered by the authors as "easy to check". If someone could explain to me why the set at ...
1
vote
0answers
72 views

The state-transition-matrix of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
2
votes
0answers
66 views

SVD when only off-diagonal terms are known

I have a real matrix $A \in \mathbb{R}^{n\times n}$ such that: $A$ is symmetric All the off-diagonal terms are known and positive Has rank $k<n$ Unfortunately I don't know the values of the ...
19
votes
1answer
1k views

A curious eigenvalue inequality

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...
1
vote
0answers
63 views

Does this system of equations have a closed form solution? [closed]

I am faced with the following system of equations and I'm looking for tools that allow me to characterize its solutions. The unknowns are $\{x_i\}$, $\{y_i\}$, and $\lambda$, for $i=1,...,N$. So there ...
7
votes
0answers
241 views

How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...
1
vote
3answers
129 views

Norm of an operator formed using a unitary operator

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
2
votes
1answer
93 views

Inverse Hadamard determinant inequality

As far as I remembered there is an inverse Hadamard inequality for the determinant of the form $$ |D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)} $$ providing all values in $(\cdot)>0$. ...
8
votes
0answers
232 views

Conjecture on matrix with reciprocal principal minors

Some notation: $A(\alpha|\beta)$ is the submatrix of $A \in \mathbb{R}^{n \times n}$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the ...
0
votes
0answers
68 views

Optimization with vectors

I am trying to solve the following optimization problem as a small part of a research project, and I do not know if there exists closed form solutions. My linear algebra is very rusty and I am looking ...
1
vote
1answer
65 views

Inverting two paraboloid relations

Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations: $$ k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right| $$ $$ ...
7
votes
1answer
191 views

Hopfian modules

My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
0
votes
1answer
95 views

Positive definite - Inverse of sparse symmetric matrix

Consider a matrix $P\in \mathbb{R}^{n\times n}$ such that at most $m<n$ elements of each column are non zero and $P$ is symmetric. I would like to find the sufficient condition(s) such that $P^{-1}...
4
votes
0answers
137 views

Closed-form solution of a linear programming question

Among all the probability matrices \begin{equation*} P = \left(\begin{array}{cccc} p_{00} & p_{01} & \ldots & p_{0,J-1} \\ p_{10} & p_{11} & \ldots & p_{1,J-1} \\ \vdots & \...
3
votes
1answer
82 views

Finding eigenvalues of an 'almost-tridiagonal' circulant matrix

Consider the $2N\times 2N$ matrix $$A=\begin{pmatrix} a &1 &0&0&0&\ldots&0&1 \\1 &-a&1 & 0 &0 & \ldots & 0&0 \\0 &1&a&1&0 &...
0
votes
1answer
43 views

The effect of linear transformation on generic vectors [closed]

I have a question about the effect of applying a linear transformation $M$ in $\mathbb{R}^{n \times n}$ to a vector $v \in \mathbb{R^n}$. I know that if $M$ has p-norm $\|M\|_p = \lambda$, then by ...
3
votes
1answer
112 views

Two minimization problems using singular value decomposition

Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are $L^...
2
votes
1answer
114 views

Commuting nilpotent matrix collection

For every large enough $m\in\Bbb N$ are there $c=\alpha m$ (for some fixed $\alpha>0$) square matrices $A_1,\dots,A_c$ that commute with each other with nonzero product ($\forall i,j\in\{1,\dots,t\}...
8
votes
1answer
140 views

Exact eigenvalues of a specific tridiagonal matrix

I'm studying the following tri-diagonal matrix $$ X = \begin{pmatrix} 0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\ x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
4
votes
1answer
125 views

Representing a graph's vertices as linear combinations of paths

I have a (directed graded) graph $G=(V,E)$ with two distinguished subsets of $V$: sources and destinations. I am to show that the set of indicator functions of paths starting in a source and ending in ...
0
votes
0answers
175 views

A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration $\hat{c} = f(c)$, where each element of $\hat{c}$ is given by $$\hat{c}_{\ell} = \frac{\sum_k Y_{k,\ell}\frac{1}{|c_{\ell}|^2+|c_{k}|...
9
votes
1answer
266 views

How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently

Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute $$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$ One option is to simply ...
3
votes
0answers
63 views

How to construct large sets of $m$-dimensional vectors over a finite field such that any $m$ of them are independent?

Let $m\ge 2$ be an integer and $\mathbb{F}$ be a finite field of order $p^k$. I want to construct as many as possible $m$-dimensional vectors $v_1,v_2,\ldots,v_n$ in field $\mathbb{F}$ such that any $...
2
votes
0answers
84 views

Quantum Groups and quantum spaces - From algebra to Analysis

My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic ...
1
vote
2answers
109 views

Any analysis on phase of eigenvalue of unitary matrix?

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis ...
0
votes
0answers
52 views

Separable Least squares - is there a notion of conjugate directions?

I have a general question. Suppose I have the following to optimize $$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$ where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...
9
votes
2answers
508 views

Do smooth manifolds admit linear atlases? [duplicate]

There is a theorem of Whitney showing that a smooth manifold can be endowed with a compatible real-analytic atlas (later, it was proven that this analytic structure is essentially unique). I am ...
2
votes
2answers
440 views

For a ring R, does $GL_n(R)$ embed into $GL_m(F)$ for some field F?

Suppose $R$ is a noncommutative ring. What is the sufficient and necessary condition for $GL_n(R)$ embedding into $GL_m(F)$ for some field $F$? In particular, what if $R$ is the group ring $\mathbb{Z}...
0
votes
2answers
499 views

Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...
0
votes
1answer
84 views

Large Tridiagonal Matrix - Eigenvalues

Consider large tridiagonal matrix (where $a$ and $b$ are real numbers): $M = \begin{pmatrix} a^2 & b & 0 & 0 & \cdots \\ b & (a+1)^2 & b & 0 & \cdots & \\ ...
0
votes
1answer
166 views

$P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable. $Z$ is eigenvalue of $P(Z )$ if $\...
1
vote
1answer
45 views

On optimizing a function whose projection and projected vector go through a linear transformation

Assume the two sets of vectors $\{\mathbf{a}_1,\ldots,\mathbf{a}_N\}$ and $\{\mathbf{b}_1,\ldots,\mathbf{b}_N\}$ of equal length. My goal is to find the optimum matrix $\mathbf{C}$ to the following ...
7
votes
1answer
143 views

Can every trace preserving isomorphism of unital self-adjoint matrix algebras be realized as conjugation by a unitary?

In this paper, Friedland shows (in Lemma 3.4) that if $\phi$ is an isomorphism of coherent algebras, then there exists a unitary $U$ such that $$ \phi(M) = UMU^\dagger$$ for all $M$. I am wondering if ...