Tagged Questions

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

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-3
votes
0answers
42 views

Help with determining onto (surjective) [on hold]

The question is to determine if the following function T(x,y,z) = (ysin x,zcos y,xy) is onto. So far I have only learned of creating a coefficient matrix and checking if the determinant is 0 to figure ...
0
votes
0answers
95 views

Problem regarding orthogonal vectors

Suppose $C_{1},C_{2},...,C_{n}$ are $0-1$ vectors of length $m$. Given $C_{i} \in \{0,1\}^{m}$ with $C_{i}=x_{i1}x_{i2}...x_{im}$ we say $C_{i}'=x_{i1}'x_{i2}'...x_{im}'$ is a subvector of $C_{i}'$ if ...
-3
votes
0answers
50 views

Rank- nullity theorem of linear transformations. Help! [on hold]

I have tried my best to prove the following statement but still fail to do so. Hope that you could help me with this. Let U, V and W be the vector spaces (over real numbers) and let S: U --> V and ...
5
votes
0answers
121 views

Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative. ...
5
votes
3answers
87 views

Minimize distance between centroids of subsets of points

In a n-dimensional space, I want to divide a set of m points into v (non-empty) subsets. I want to minimize the sum of the pairwise Euclidean distances between the centroids of the resulting subsets. ...
5
votes
1answer
154 views

Anti-bidiagonal matrix with main anti-diagonal {1,2,3,…} and first sub-anti-diagonal {-1,-2,-3,…} has eigenvalues lambda={1,-2,3,-4,…}

Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows $$ B_6=\begin{bmatrix} & & & & & 6\\ & & & ...
4
votes
0answers
137 views

Proving that the kernel of this matrix is of dimension 2

Using a computational software program, I found that the kernel of the following matrix is of dimension 2 but I haven't managed to prove it: \begin{equation} \text{for almost all } t_1>0,\quad ...
4
votes
2answers
158 views

Do singular values dominate eigenvalues?

Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that $|\lambda_1| \ge |\lambda_2| \ge \cdots ...
0
votes
0answers
76 views

Solving matrix equation (AX)^2+(BY)^2=D [closed]

Is there any method that can solve the matrix equation in such a form (AX)^2+(BY)^2=D? A and B are matrix, X, Y and D are column vectors. (Solve for X and Y) I originally have two equations such as ...
1
vote
2answers
139 views

Question on Posets and open sets [on hold]

i'm sorry if my question is really trivial but this one is really bugging me out.. So let's have a partially ordered set $I$ with the topology in which the open sets are the increasing ones: $i\in U$ ...
7
votes
2answers
174 views

Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that: The entries of $A$ are $\in \{0, 1\}$. For all pairs of columns $u, v$ of $A$ the entries of $u - ...
0
votes
0answers
19 views

Comparing the inverse of a diagonally dominant matrix [migrated]

I have a $d \times d$ matrix $A$ whose entries are bounded (C1): $I - \epsilon X \preceq A \preceq I + \epsilon X$, where $I$ is the identity matrix and $X = 11^\top - I$ is the matrix with ones ...
0
votes
0answers
31 views

Questions about some special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
1
vote
1answer
142 views

Is the notation ${}^t g$ for the transpose of a linear transformation intended to be suggestive?

The notation ${}^t g$ for the transpose of a linear transformation is, in my view, quite unusual: otherwise (at least in many areas of math), one almost never sees subscripts or superscripts appearing ...
2
votes
1answer
100 views

Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?

The nonnegative matrix $V = \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right)$ has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = ...
1
vote
0answers
51 views

Is there an efficient way to compute the “complete subset regression”?

Background: Let $X \in \mathbb{R}^{N\times K}$ and $y \in \mathbb{R}^{N\times 1}$ be data for a regression problem. The aim is to find $\beta \in \mathbb{R}^{K\times 1}$ such that $X\beta \approx y$ ...
2
votes
1answer
72 views

Linearly constrained eigenvalue problem

Suppose I'd like to: \begin{align} \mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\ && ...
1
vote
0answers
13 views

How to restructure adjacency matrix $A$ from shortest distance matrix $B$ in Network topology inference

An undirected graph with $n$ nodes could be referred to as an adjacency matrix $A$. $A=[a_{ij}]_{n×n}$ with $a_{ij}=a_{ji}=1$ standing for there being an edge between node $i$ and node $j$, and no ...
4
votes
1answer
95 views

Writing a complex orthogonal matrix as a conjugation by real orthogonal matrices

Let $Q\in O(n,\mathbb C)$ be a complex orthogonal matrix. I would like to know if $Q$ can always be written as $Q = T^{-1}ST$, where $T\in O(n,\mathbb R)\subset O(n,\mathbb C)$ and $S$ belongs to some ...
1
vote
1answer
51 views

Looking for algorithms based on sorting [closed]

i am looking for algorithms which use sorting in low-dimensional space like $R$ and how they are generalized for higher-dimensional spaces like $R^2$ where there is no sorting possible. (i.e. numbers ...
3
votes
2answers
87 views

eigenvalue estimate of the adjacency matrix

The adjacency matrix of a nonempty (undirected) graph has a strictly positive largest eigenvalue $\lambda_\max$. A very easy upper estimate for it can be obtained directly by Gershgorin's theorem: $$ ...
0
votes
1answer
251 views

Is there a Gröbner basis analogue that exists for vector spaces?

Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$. ...
0
votes
2answers
97 views

Can any antilinear involution be trivialized by a change of basis?

Consider an antilinear involution, that is an antilinear map on a complex vector space, whose matrix $M$ obeys $MM^*=1$ where the star denotes complex conjugation. Can we find a change of basis whose ...
4
votes
1answer
212 views

Generalized Cauchy-Binet sum over a fixed subset of indices

I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
3
votes
2answers
136 views

Linearly independent family of sequences of rationals with a cardinal equal to the continuum

I'm coming back to this question. Is it possible to have "an explicit" linearly independent family of sequences of rationals with a cardinal equal to the continuum? PS: sorry for the duplicate on the ...
0
votes
0answers
55 views

The space of sequences of rationals and its dimension [duplicate]

In the following page, I give an example of a vector space not isomorphic to its double dual. I use the space $E$ of sequences of reals. Its dimension (over the field of the reals) is the one of the ...
0
votes
0answers
15 views

Significance of Eigenvectors of a Covariance matrix [migrated]

In PCA and in many other problems of machine learning we use Eigenvectors of covariance matrix of the data. How do we visualize Eigenvectors of Covariance matrix? The Principal Eigenvector ...
2
votes
0answers
35 views

Random matrices whose limit gives exact Wigner surmise

Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write ...
1
vote
0answers
73 views

MInors related problem [closed]

A matrix $A$ has $m$ rows and $n$ colums, such that $m \leq n$. We know that each row of $A$ has the norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
5
votes
3answers
199 views

Why is the spectrum of this matrix product invariant with respect to order of the multiplicants?

I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A_1,...,A_k \in \mathbb R^{n\times n}$ and I'm observing that the ...
11
votes
1answer
180 views

Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...
2
votes
0answers
152 views

Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$ Let me give a ...
7
votes
1answer
138 views

Compute only selected components of an eigenvector

I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...
4
votes
1answer
170 views

$n$ columns of a specific “infinite” Vandermonde matrix always linearly independent? [closed]

Consider the "infinite" Vandermonde matrix $$ V (x_1, x_2, \ldots , x_n) = \begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\ ...
3
votes
1answer
97 views

Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$

I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...
-1
votes
1answer
81 views

Action of rotation group on Matrices [closed]

Is the following assertion true? Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in ...
-3
votes
1answer
121 views

A problem that involves matrix and Lorentz Transformation [closed]

To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes. $1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G ...
1
vote
1answer
71 views

Connection between eigenvalues of A and its LDL decomposition

Consider an undirected graph $G$ with $N$ vertices and its adjacency matrix $n_{ij}$: $n_{ij} = 1$ if vertices $i$ and $j$ are connected by an edge and $n_{ij} = 0$ otherwise. Consider $A_{ij} \equiv ...
1
vote
2answers
107 views

The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...
3
votes
0answers
58 views

Dimension of the sum of images of transpose

$\newcommand{\rank}{\operatorname{rank}}\newcommand{\im}{\operatorname{im}}$ Given $A,B\in M_{n\times n}(k)$, define $\rank(A,B):=\dim(\im A+\im B)$. I'm looking for results regarding relationships ...
9
votes
1answer
237 views

What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are individual eigenvectors ...
2
votes
1answer
70 views

Solution to generalized Sylvester equation

I am interested in solving generalized Sylvester equations (for $X$) of the form: $$ \sum_{j=1}^k A_j X B_j^T = F, $$ where $A_j,B_j,X,F\in\mathbb{C}^{n\times n}$ and $k$, $n$ are integers. I will ...
1
vote
0answers
79 views

Reference request on operator matrices [closed]

I'm looking for a reference on linear, bounded, self-adjoint operators defined on the product space, $T:E\times F\to E\times F$ such that $$Tx = \begin{pmatrix}A & B \\ C & D ...
0
votes
1answer
95 views

Symmetric Zero-Diagonal Matrices

Consider matrices with entries in a field $F$ of characteristic $2$. Let $\Omega$ denote the $2n\times2n$ matrix $\left[\begin{array}{ll}0&1_n\\1_n&0\end{array}\right]$. Then $X^t\Omega X$ is ...
0
votes
0answers
59 views

Speed up Linear programming

I have a linear programming problem like this: minimize $c^t X$ under the constraint that $AX \ge b$. I will need to solve this linear programming problem online many times. I need it to be as fast ...
0
votes
0answers
26 views

Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space. My goal is to find all the ...
1
vote
1answer
96 views

Checking the intersection of two sets

Let $E\subset{\mathbb R}^n$ be a set of the type $I_1\times \dots \times I_n$, where $I_k$ are real intervals, and $X$ be and $n\times p$ real matrix. Suppose also that $rank(X)=p$ and $n>p$. Is ...
0
votes
0answers
37 views

Multiplicative Symmetrization of Bilinear Forms

Let $F$ be a field of characteristic $2$, let $V$ be an even dimensional $F$-vector space, let $B$ be a non-degenerate symplectic bilinear form on $V$ and let $^*$ be the adjoint of $B$ on ${\rm ...
0
votes
0answers
48 views

Applying a linear operator to a basis set following SVD orthonormalization

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases. My intention is to: decompose $\Phi$ via SVD: $U \Lambda V^T = \Phi$ to create it's ...
1
vote
0answers
112 views

Bounds on the effect of a matrix product on the Frobenius norm

I was wondering if there was a way to put upper and lower bounds on the Frobenius norm of a matrix product in relation to the Frobenios norm of one of the individual matrices, i.e, ...