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38 votes
6 answers
11k views

Is there a version of inclusion/exclusion for vector spaces?

I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ...
1 vote
1 answer
391 views

How one can show that this matrix is full rank?

Fix $d\in\mathbb{N}$ and consider $e_{i,j}\in\mathbb{C}$ for $i=1,\dots,d+3$ and $j=1,\dots,d-1$. Suppose to have the following matrices $$N_{i,1}=\begin{pmatrix} 1 & 0 \\ e_{i,1} & 1 \end{...
5 votes
1 answer
429 views

Lower bound for the rank of the sum of $n$ matrices

I found a mathematical note by George Marsaglia entitiled "Bounds for the rank of the sum of two matrices", where he proves the following result. Let $A_1$ and $A_2$ be two complex matrices ...
3 votes
1 answer
189 views

Rank properties of matrix valued in linear forms

Let $R(X,Y) \in \text{Mat}_{d,d+1}(\mathbb{C}[X,Y]_{(1)})$ be a $d \times (d+1)$-matrix valued in linear forms $\mathbb{C}[X,Y]_{(1)}:= \{aX+bY \ \vert \ a,b \in \Bbb C \}$. Let denote $v_j(X,Y)$ its $...
3 votes
1 answer
99 views

Eigenvectors of $P^\top P$ for 0/1 matrices $P$

Let $P$ be an $m \times N$ matrix of zeros and ones (think $N \gg m$), and let $\mathbf{u} \in \mathbb{R}^N$ be a unit vector satisfying $P^\top P\mathbf{u} = \lambda^2 \mathbf{u}$ for some $\lambda &...
5 votes
1 answer
148 views

A general form of a maximal totally isotropic subspace in the split octonion algebra

$\DeclareMathOperator\Ker{Ker}$Let $\mathbb O'$ be the split octonion algebra over $\mathbb R$. For each nonzero divisor of zero $x\in \mathbb O'$ ($x \neq 0$, $N(x)=0$) the kernel of the left ...
1 vote
0 answers
175 views

Solution of recurrence relation with summation

I have the following recurrence relation: $$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
3 votes
1 answer
252 views

Two isotropic subspaces in a symplectic vector space

Let $k$ be a field of characteristic $0$, let $V$ be a finite-dimensional vector space over $V$, and let $\omega(-,-)$ be a symplectic bilinear form on $V$. In other words, $\omega(-,-)$ is an ...
9 votes
3 answers
729 views

Math history research: a copy of "Zur relativen Wertbemessung der Turnierresultate" , eigenvector centrality by Edmund Landau

I'm searching for a copy of an old paper made by Edmund Landau: Zur relativen Wertbemessung der Turnierresultate, Deutsches Wochenschach, 11. Jahrgang (1895), 366–369. However, I can't find it ...
0 votes
0 answers
43 views

Absolute value of elements of b=Ax and the minimum singular value of A

For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_\text{min}$, of $A$? What I want is something like: $\sigma_\...
2 votes
0 answers
75 views

Smallest dimension, on which a set of matrices acts non-trivially

Let $A_i$, $i=1,\dots,N$, be a finite set of $D<\infty$ dimensional Hermitian matrices. Let $d$ be the smallest number for which there exists a unitary $D$-dimensional matrix $U$, and Hermitian $d$-...
9 votes
1 answer
845 views

Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions

I asked this question on MSE here. Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
5 votes
1 answer
279 views

Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?

Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $: $$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
0 votes
0 answers
40 views

Eigendecomposition of Toeplitz matrix

I am working with Toeplitz matrix, I know that a Toeplitz Matrix $T$ can be decompose as a sum of a circulant and skew circulant matrix which can be diagonalized using the DFT matrix $T=C+S = F\...
3 votes
1 answer
196 views

Deriving the "Explicit" formula for inverse of Hilbert/Cauchy matrices

My exact question is, how to derive the formula for $H^{-1}$, in which $H_{ij}=\frac{1}{i+j-1}$. I am currently working my way through Hoffman&Kunze Linear Algebra. I noticed that a question on ...
2 votes
2 answers
380 views

(Non-topological) interior of a convex set

Given a convex subset $X$ of a real vector space $V$, I'm interested in the set $$Y:=\{x\in X:\ \forall v\in V, \ \exists\epsilon>0 \text{ s.t. } x+\epsilon v \in Y \}.$$ My question is boring: ...
1 vote
0 answers
139 views

Integral convex polytopes formed from the weight diagrams of representations of $\mathfrak {sl}_4$($\mathbb{C}$)

I'm a student studying undergraduate abstract algebra and doing a summer research project in the mathematics department at my school. I'm barely familiar with the rudiments of representation theory; I ...
20 votes
6 answers
42k views

Eigenvalues of symmetric tridiagonal matrices

Suppose I have the symmetric tridiagonal matrix: $$ \begin{pmatrix} a & b_{1} & 0 & ... & 0 \\\ b_{1} & a & b_{2} & \ddots & \vdots \\\ 0 & b_{2} & a & \...
0 votes
0 answers
129 views

Linearly independent Kronecker product construction

I have a question regarding a constructive argument about Kronecker products which came up while trying to solve a more general problem. Let $n\in \mathbb{N}$ and $E \subseteq [n] \times [n]$ with $d^...
1 vote
0 answers
39 views

Constructing a centered distribution absolutely continuous with respect to uniform measure on the sphere with a pre-specified covariance?

Let $\mathcal{K}_n$ denote the space of $n \times n$, real symmetric positive semidefinite matrices $K$ having unit trace. It is easy to verify that for each $K \in \mathcal{K}_n$, there exists a ...
3 votes
0 answers
109 views

How much a general a theory of matrices equivalence under group actions we have?

Let $F$ be a field and let $M_{m,n}\,(F)$ be the $F$-linear space of $m \times n$ matrices over $F$. Let $G$ be a group acting on $M_{m,n}\,(F)$. My question is: Do we have some theory about the ...
0 votes
0 answers
101 views

Eigenvectors of tridiagonal hermitian matrix

In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as: \begin{align*} Q_n = \begin{pmatrix} 0 & q_{1,2} & 0 & 0 & ...
0 votes
0 answers
32 views

Finding measure representation for rank 2 moment matrices

Assuming the following equation has a solution, I'm interested in finding any concrete values of $x_{1},\dots x_{n},y_{1},\dots y_{n},c_{1},c_{2},R$ that fulfills it. $$ \begin{bmatrix} 1 & 1 \\ ...
3 votes
1 answer
102 views

Literature containing basic knowledge of homogeneous functions

Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(...
7 votes
1 answer
388 views

Questions on symmetric Hadamard matrices

Definitions: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal. If $A$ is a symmetric matrix, then $A = A^T$ and if $...
2 votes
1 answer
237 views

Geometric interpretation of trace of a linear operator

This question is really an addendum to Geometric interpretation of trace There is a nice account of the trace in Chris Doran's thesis here: http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/...
7 votes
2 answers
201 views

When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?

Let $M_n(\mathbb{C})$ represent the space of $n \times n$ matrices over $\mathbb{C}$. We will think of it as a $\mathbb{C}$-vector space. Notice that if $A \in M_n(\mathbb{C})$ is invertible, then the ...
368 votes
31 answers
80k views

Geometric interpretation of trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandary; Is there a geometric interpretation of the trace of a matrix? This question ...
7 votes
1 answer
271 views

Existence of a linear map resulting in the determinant being an elementary symmetric polynomial

Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
9 votes
1 answer
563 views

Peter–Weyl decomposition of a group representation rather than group algebra

Consider a finite or compact group $G$. The Peter–Weyl decomposition is usually formulated for the group algebra $\mathbb{C}[G]\simeq\bigoplus_i \operatorname{End}(V_i)$, where $V_i$ are the spaces of ...
7 votes
3 answers
5k views

Proof for a rank-one decomposition theorem of positive (semi) definite matrices

$\DeclareMathOperator\rank{rank}\DeclareMathOperator\trace{trace}$Consider the following result which I recently came across in a research paper in my area (signal processing) Let $X$ be a $N\times N$...
7 votes
2 answers
1k views

Why is the spectrum of Erdős–Renyi random graph approximately symmetric?

I am recently self-learning random matrix theory and made some simulations about the spectrum of Erdős–Renyi random graph $G(n,p)$ when $np\to\infty$, and $np\to c=2,3$. The plots above are already ...
9 votes
2 answers
245 views

Matrix invariants for simultaneous conjugation by a finite subgroup of $\textrm{GL}_n$

Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ ...
0 votes
1 answer
171 views

Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$

Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group $G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
1 vote
1 answer
114 views

Sum of squares of $k\times k$ cofactors is $1$ for an orthonormal matrix [closed]

Let $n,k\in \mathbb N$ with $k\leq n$. Let $A$ be an $n\times n$ real orthonormal matrix. Fix any $k$ rows of $A$ and from there consider every possible $k\times k$ cofactors and there will be exactly ...
0 votes
1 answer
269 views

Product of subspace and its inverse

$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
7 votes
2 answers
603 views

Minimizing the largest eigenvalue of random matrices

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix with entries $A_{ij} \sim \mathcal{N} (0,1)$, all independent except for the symmetry condition. Consider the following minimization problem:...
-1 votes
1 answer
825 views

How to calculate determinants of such types?

Consider next determinant that we want to expand around $h=1$ \begin{eqnarray} Z_q \ = \ h^{N N_f} \ \ \left ( \prod_{n=1}^{N} \ \sum_{l_n=0}^{N_f -q} \ h^{2l_n+q} \ \binom{N_f}{l_n} \right ) \ \...
4 votes
1 answer
314 views

Some but not all eigenvectors mutually orthogonal

Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are ...
0 votes
0 answers
77 views

Eigenvalues of N×N correlation matrices as N tends to infinity

I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero. ...
1 vote
0 answers
110 views

Multiplication of a matrix by sub-matrices

I have a $J \times J$ matrix $C$ that is upper triangular. Also, $C'C$ is positive definite. I also have a matrix $A$ formed by submatrices of size $J \times K$ as follows $$ A = \begin{bmatrix} A_1 \\...
0 votes
0 answers
109 views

Linear independence in $\mathbb{Z}_q^n$

Consider $\mathbb{Z}_q \equiv \mathbb{Z}/q\mathbb{Z}$, where $q \geqslant 2$. A set of vectors in $\mathbb{Z}_q^n$ is said to be linearly independent if no nontrivial linear combination of them ...
4 votes
1 answer
685 views

Who and when proved Artin's Theorem on alternative rings?

I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings). Question. When has Artin proved this theorem and where was it published for the first ...
2 votes
0 answers
160 views

An "almost" true inequality for Hermitian matrices

Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
1 vote
1 answer
452 views

About the Hadamard conjecture

On the wikipedia article about Hadamard Matrix it says that "The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is $92$" But it also says that ...
5 votes
0 answers
583 views

Dimension inequality for subspaces in field extensions

Let $K\subset L$ be a field extension and $A, B\subset L$ be $K$-subspaces of $L$ of finite positive dimensions. Assume further that for every $a, b \in L$ and every nontrivial proper finite ...
6 votes
0 answers
130 views

Bent vectors and $\pm 1$ eigenvectors with respect to non-Sylvester Hadamard matrices

A Hadamard matrix is an $n\times n$-matrix $H$ where each entry in $H$ is $\pm 1$ and where $H/\sqrt{n}$ is orthogonal. It is well-known that if $H$ is an $n\times n$-Hadamard matrix, then $n<3$ or ...
3 votes
2 answers
251 views

Minimum-norm solution of $A = X B + B^T X^T$

Let $A, B, X$ be invertible square matrices, and let $A$ additionally be symmetric. I'd like to solve the following minimization problem: $$\text{argmin}_X |\!| X |\!|_F \ \ \ \ \text{s.t.} \ \ \ \ A =...
3 votes
4 answers
551 views

How big a class of lines can a non-linear transformation map to itself?

Edit: In the original version of this question, I wrote "lines through the origin" instead of "lines"; as Alexandre Eremenko points out in his answer, this makes the question too ...
2 votes
2 answers
286 views

Inequality on numerical range of inverse of kernel matrix

Let $k(.,.)$ be a function that takes two vectors as input and outputs a scalar as follows \begin{align} \mathcal{k}(x,y) = \exp\left(-\frac{\|x-y\|_2^2}{2}\right) \end{align} where $\|x\|_2$ denotes ...

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