Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

learn more… | top users | synonyms

28
votes
0answers
2k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
13
votes
0answers
252 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
11
votes
0answers
227 views

Minimal and characteristic polynomials on tensor product spaces

Given two finite-dimensional vector spaces $V$ and $W$ over a common field $k$ as well as $k$-linear transformations $\varphi \colon V \to V$ and $\psi \colon W \to W$, what can be said in general ...
11
votes
0answers
356 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= \begin{bmatrix} 1 ...
11
votes
0answers
493 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
10
votes
0answers
147 views

On the maximum number of polynomials in a certain subspace

I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that. Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with ...
9
votes
0answers
101 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
9
votes
0answers
179 views

Mathematical properties of two dimensional projection of three dimensional rotated object

Please be gentle as I do not have any degree in maths. By using a compass/straighedge method to construct Metatron's cube, a regular dodecahedron can be inferred from intersecting points. I'm looking ...
9
votes
0answers
129 views

What does the space of non-diagonalizable matrices look like?

Let $k$ be a field (I would be happy working entirely over $\mathbb C$). Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices ...
9
votes
0answers
482 views

Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices

The component-wise product (Hadamard product) of two positive definite matrices is a positive definite matrix (Schur product theorem). I encountered the following proof of it: $A=(a_{ij})$ and ...
9
votes
0answers
244 views

How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?

I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66. ...
9
votes
0answers
3k views

Inverse of a Toeplitz Matrix

A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is an $n\times n$ Toeplitz matrix: $$ A ...
8
votes
0answers
302 views

How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
8
votes
0answers
108 views

Linear functional equation

During my mathematical musings I encountered the following functional equation : denote by $L$ the set of all functions ${\mathbb Z}^2 \to {\mathbb C}$ satisfying $$ \begin{array}{cl} ...
8
votes
0answers
221 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
8
votes
0answers
558 views

Eigenvalues of doubly stochastic matrices

There was a long standing conjecture stating that the geometric location of eigenvalues of doubly stochastic matrices of order $n$ is exactly the union of regular $k$-gons anchored at $1$ in the unit ...
7
votes
0answers
161 views

Here is a riddle that I have no idea how to solve.

Okay, so I was trying to solve this riddle found here. It is a diagram of a star with 16 points. Each point corresponds uniquely to a number between 1 and 16. The letters on each point represent a ...
7
votes
0answers
83 views

What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
7
votes
0answers
196 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix. I have already asked a (viewed but unanswered) ...
7
votes
0answers
98 views

A very difficult problem about the existence of following $SU(2)$ matrices?

Let $G_i$ be a sequence of $2\times2$ $SU(2)$ matrices, where $i=1,2,...,n$; and $P$ represents a permutation of $\left \{ 1,2,...,n \right \}$. The question is: Does there exist a sequence of ...
7
votes
0answers
220 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
7
votes
0answers
98 views

Maximum determinant of latin squares

I strongly conjecture that the maximum absolute determinant of a latin square can be attained by a circulant matrix. For example, $\pmatrix {5&4&2&3&1 \\ 1&5&4&2&3 \\ ...
7
votes
0answers
164 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
7
votes
0answers
389 views

Monotone matrix norms

[Ciarlet 2.2-10] Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if ...
7
votes
0answers
478 views

Condition of an eigenvector problem

Please, somebody help me with this problem. [Ciarlet 2.3-5] Let ${A}$ and ${B} = {A} + \delta{A}$ be two symmetric matrices with eigenvalues $$\alpha_1\ \leq\ \alpha_2\ \leq\ \ldots\ \leq\ ...
7
votes
0answers
307 views

Computing the SVD factorization on C++ (using the proof of the existence of the SVD factorization)

I am doing a C++ program that computes the SVD factorization of a real matrix A without using any known library of algebra that contains the implementation. In addition, QR descomposition is not ...
7
votes
0answers
179 views

Reference suggestion: eigenvalues of tridiagonal matrices

I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals). I have seen authors use continued ...
7
votes
0answers
123 views

Continuous choice of basis for subspaces

Consider the flag variety (or flag manifold, depending on who you are) $V=\mathrm {Fl} (3,\mathbb C)$ of complete flags of subspaces of $\mathbb C^3$. That is, an element of M is a tuple (L , P) ...
7
votes
0answers
390 views

Pfaffian properties

Given a $2n\times 2n$ real skew-symmetric matrix $A$, its Pfaffian $\mathrm{Pf}$ is defined as: $$ \mathrm{Pf}(A) = \frac1{2^n n!}\sum_{\sigma\in S_{2n}}\mathrm{sgn}(\sigma)\prod_{i=1}^n ...
6
votes
0answers
64 views

Is there a mathematical way to fold a $20 dollar bill for compactness?

I had a strange thought. I used to carry a pill fob on my keys with an emergency $20 bill in it, before the whole thing got stolen. I always had some trouble fitting the bill inside the fob and ...
6
votes
0answers
52 views

Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
6
votes
0answers
71 views

Reference for Cavalieri's principle

Does someone know of a reference where I can see Cavalieri's principle (basically the principle that generalized areas can be obtained by multiplying "base times height" -- for constant ...
6
votes
0answers
96 views

Geometric interpretation of $x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots$

Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product: $$x_1y_1+x_2y_2+x_3y_3+\dots+x_ny_n$$ Is the cosine of the angle between them. For an application I was ...
6
votes
0answers
130 views

Exploiting structure in multilinear equations

I'm wondering if there are any standard techniques for exploiting structure in multilinear equations. An example of what I have in mind is solving $A_{ab} X_{bc} A_{cd} (B_{ad} B_{bc} + B_{ac} ...
6
votes
0answers
249 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
6
votes
0answers
105 views

A power of the characteristic polynomial

Let $A$ be a square matrix with real or complex coefficients of size $n$. Define its characteristic polynomial by $\chi_A(X) = \det(A-XI_n)$ (or $\det(XI_n-A)$ if you prefer). The question is : Prove ...
6
votes
0answers
79 views

Strength of “Every finite dimensional subspace of a vector space has a complement”

Does the following choice principle have a name? Every finite dimensional subspace of a vector space has a complement. Equivalently, every line inside a vector space has a complementary ...
6
votes
0answers
182 views

Set geometry and inclusion

I would like to prove that the set of the symmetric positive semi-definite matrices which is defined as $$\Delta_2= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad ...
6
votes
0answers
221 views

Holder inequality for matrices

I am interested in the following version of the Holder inequality. Let $D \in M_n(\mathbb{C})$ be a positive semi-definite matrix of trace $1$ and $A, B \in M_n(\mathbb{C}).$ Does it follow that $$ ...
6
votes
0answers
183 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
6
votes
0answers
341 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
6
votes
0answers
1k views

Tensor Product is associative, distributive, not commutative.

Tensor Product is associative, distributive, not commutative. Here is my attempt to show tensor product is associative, is it legit? If $T$ is a $p$-tensor and $S$ a $q$ tensor, then $T \otimes ...
6
votes
0answers
90 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
6
votes
0answers
869 views

Farkas Lemma proof

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
6
votes
0answers
440 views

Generalized Eigenvalue Problem with one matrix having low rank

I have a specific Generalized Eigenvalue Problem (GEVP) where i am primary not interested in solving this problem but concluding from a standard EVP the spectrum of the GEVP. The Problem Let $A$ be ...
6
votes
0answers
127 views

Piecewise Affine Bijections of $\mathbb{R}^n$

I have a min-max function $f:\mathbb{R}^n\to\mathbb{R}^n$ of the form $$f(x) = \min_{i=1,\dots,n}\max_{j=1,\dots,n}(\alpha_{ij}^Tx + \beta_{ij})\quad\text{where each } \alpha_{ij}\in ...
6
votes
0answers
555 views

Is every convex-linear map an affine map?

Let's say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$. ...
6
votes
0answers
205 views

Is there a Mazur–Ulam theorem equivalent for vector spaces over finite fields?

I know that Mazur–Ulam theorem holds for normed linear spaces over $\mathbb{R}$. I wanted to know whether under some "weak" conditions on the map $f$, can we have Mazur-Ulam theorem for a vector ...
6
votes
0answers
220 views

Uses of Chevalley-Warning

In the recent IMC 2011, the last problem of the 1st day (no. 5, the hardest of that day) was as follows: We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The problem asks : Prove the ...
6
votes
0answers
480 views

Hardness of finding eigenvalues over finite fields

How hard is it (computationally) to find eigenvalues/eigenvectors of matrices over finite fields? Suppose the field has size exponential in the input. (Does the QR algorithm still converge?) How ...