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Questions tagged [linear-algebra]

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

2
votes
2answers
51 views

Rate of convergence for eigendecomposition

Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta_D$ on a set of cardinaltity 4 is the matrix $$\Delta_D := \left( \begin{matrix} 2 &...
-1
votes
0answers
27 views

Understanding Partial Derivatives of a Neural Network

I have to compute the following double derivative: $$ \partial _{x_i} \nabla_W \sigma(f(W,x))$$ where $W = (W_1, W_2, \dots, W_L)$ is the set of weight matrices, $f(W,x)$ is a $linear$ neural ...
0
votes
0answers
15 views

Solution to quadratically constrained QP as linear combination of eigenvectors

Let $\textbf{M}$ be a symmetric $n\times n$ matrix with zero diagonal, the strong Frobenius property, and spectral radis $\rho(\textbf{M}) <1$. Define $\textbf{R} := (\textbf{I} - \textbf{M})^{-2}$,...
2
votes
1answer
53 views

Bounding entries of the inverse of a matrix with bounded entries

Let $A$ be an $n$-by-$n$ matrix with integer entries whose absolute values are bounded by a constant $C$. It is well-known that the entries of the inverse $A^{-1}$ can grow exponentially on $n$. (See ...
1
vote
0answers
50 views

Infinitesimal matrix rotation towards orthogonality

TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal. Setting In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...
1
vote
0answers
63 views

Infinite products from the fake Laver tables-Now with no set theory

We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have $2^{n}*_{n}x=x$, $x*_{n}1=x+1\mod 2^{n}$,...
0
votes
0answers
108 views

How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions ...
8
votes
2answers
354 views

Constructivist defininition of linear subspaces of $\mathbb{Q}^n$?

Let me preface this by saying I'm not someone who has every studied mathematical logic or philosophy of math, so I may be mangling terminology here (and the title is a little tongue in cheek). I (and ...
0
votes
0answers
44 views

Tensor factorizations [on hold]

In matrix level, if I have a parameter matrix $M$ with a shape of $d \times d$. In order to reduce parameter size, the matrix can be factorized into two matrix $A,B$ with shapes of $d \times k,k\times ...
0
votes
0answers
41 views

The derivative in inverse matrix [on hold]

I wonder how to calculate the following derivative w.r.t to matrix: $\frac{d(x^TW^{-T}W^{-1}x)}{dW}$, where $W$ is a $\mathbb{R}^{d\times d}$ matrix and $x$ is a $\mathbb{R}^d$ vector. The result ...
1
vote
0answers
33 views

Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix

Lets assume we have the following equation: $AU=\lambda U \Rightarrow\left[ \begin{array}{c|c|c} 0 &A_{12}&A_{13}\\ \hline A_{21}& 0& A_{23}\\ \hline A_{31}&A_{32}&0 \end{...
6
votes
0answers
177 views
+200

On a problem for determinants associated to Cartan matrices of certain algebras

This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested ...
9
votes
0answers
149 views

More mysterious properties of Gram matrix

This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question. The following fact could be extracted from 0402087: For any $a_i\...
3
votes
1answer
115 views

Determinant of an “almost cyclic” matrix

Let $n\geq 3$, let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for $\mathbf{d}\in \mathbb{C}^n$ let $\operatorname{diag}(\mathbf{d})$ be the diagonal matrix with $\...
-2
votes
0answers
42 views

How to infer information about order of matrix [closed]

A and B are matrixs of n order, we know A^2+B^2=AB, and AB-BA is invertible, please prove the number of order n is multiple of three. I know a matrix is uninvertible when its determinant is zero, but ...
5
votes
0answers
74 views

Generalized Eigen property of a matrix

Given a $n \times n$ invertible matrix $A$, I am interested in the set $$ \mathcal{S} = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}. $$ Thus for all eigenvalues $\lambda_i$, we have $\...
1
vote
1answer
308 views

Is this line of thought (using linear algebra to get number theoretic results) already being pursued in the literature?

Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
1
vote
0answers
51 views

Matrix trace minimization of quadratic and linear terms under orthogonal manifold constraints

How would one solve the following orthogonal manifold problem? $$\begin{array}{ll} \text{maximize} & \mbox{tr}(X^\top A X - X^\top B)\\ \text{subject to} & X^\top X = I\end{array}$$ where $A ...
4
votes
0answers
47 views

Lambek calculus, linear logic, and linear algebra

In his 1958 paper, The Mathematics of Sentence Structure, Joachim Lambek introduced the Lambek calculus. In modern terms, it could be understood as a syntax for biclosed monoidal categories, and he ...
6
votes
0answers
103 views

Gram matrix determinant in dimension 4 and $E_8$

Consider a determinant of a Gram matrix in dimension $4$. $$\begin{vmatrix} 1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\ -\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...
-5
votes
0answers
29 views

Problem of percentages [closed]

Can someone explain why: 100% of 10 = 10% of 100 or 78% of 5 = 5% of 78 I don't seem to understand why this relationship occurs and have tried to solve it algebraically
0
votes
1answer
83 views

Relationship between $2 \to 2$ norm and $\infty \to 2$ norm [closed]

I am wondering what are the best known relationship between $\|A\|_{2\rightarrow 2}$ and $\|A\|_{\infty\rightarrow 2}$ and how tight it is. E.g., the trivial result is that for matrix $A$ with ...
6
votes
3answers
276 views

$SO(m+1)$-equivariant maps from $S^m$ to $S^m$

Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$. Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$. Question 1: Is F the ...
2
votes
2answers
147 views

Solving diagonal simultaneous quadratic equations

A problem I am trying to solve has led to me to the following system of equations: $$A(x^2) + Bx + c = 0$$ Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...
10
votes
1answer
312 views

Is there a pattern for the irreducible factors and degrees of a characteristic polynomial?

Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
-1
votes
0answers
72 views

Non-negative irreducible matrices with random (correlated or independent) non-zero entries

Lets $M$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $M$, $\lambda$, is positive and equal to its spectral radius $\rho(M)$. Now assume the ...
-1
votes
0answers
33 views

A question on the Non-degenerated bilinear form [migrated]

Prove that $<C(G), C(G)>$ is dual system with bilinear form $$\int_G \phi(x)\psi(x)dx $$ and $\psi(x),\psi(x)\in C(G)$ so here i was trying bilinear form im not getting how to prove non-...
4
votes
0answers
40 views

Is there a fast way to compute the lowest eigenvalue of this symmetric PD matrix in this specific scenario?

Consider $$C = A^H D A + M$$ where $A$ is a $m \times m$ unitary matrix. $D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$. $M$ is a $m \times ...
0
votes
0answers
104 views

Infinite products of complex numbers or matrices arising from rank-into-rank embeddings

I wonder what kinds of closed form infinite products of matrices, elements of Banach algebras, and complex numbers arise from the rank-into-rank embeddings. Suppose that $\lambda$ is a cardinal and $...
3
votes
1answer
66 views

Gaussian expectation of outer product divided by norm (check)

I am trying to get compute at least the directional component of the following expectation, where $M$ is a symmetric, invertible, PD matrix: $$\mathbb{E}_{v \sim N(0, I)}\left[\frac{vv^T}{||Mv||_2}\...
1
vote
1answer
82 views

Matrix of powers of pairwise differences

Let $\underline{c}:=\left(c_1,\dots,c_n\right)$ be pairwise distinct complex numbers, and let $k$ be a non-negative integer. Define the matrix $A_{n,k}(\underline{c})$ to contain the $k$-th powers of ...
2
votes
1answer
84 views

Maximum number of $0$-$1$ vectors with a given rank

Let $k\ge2$. The maximum number of $0$-$1$ (column) vectors of length $2k-1$ which make a rank $k$ matrix with no zero row nor two identical rows is $2^{k-1}+1$. (The rank is over the rationals.) I ...
4
votes
1answer
188 views

Solving Linear Matrix Recurrences

Question: Are there standard techniques available for solving the following kind of linear matrix recurrence relations: $$M_1,\cdots,M_k\ \in\ \mathbb{R}^{m\times n}$$ $$ A_1,\cdots,A_k\ \...
8
votes
1answer
409 views

Axiom of choice and algebraic tensor product

The first part of the question was asked on Math-stackexchange. Let $V$, and $W$ be vector spaces. By the universal property of the tensor product, there is a canonical map from $V^*\otimes W^*$ ...
1
vote
0answers
34 views

The significant role of dual frames in the progress of Frame theory

For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$ $$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
4
votes
1answer
136 views

Inverse of a matrix and the inverse of its diagonals

While researching a question, I faced with the following problem: I have to prove that for a positive definite matrix we have $${\mathbf n}^T {\mathbf R}^{-1}{\mathbf n}\geq {\mathbf n}^T {\mathbf D}...
3
votes
1answer
111 views

Representation of Subgraph Counts using Polynomial of Adjacency Matrix

We consider a graph $G$ of size $d$ with adjacency matrix $A$, whose entries take value in $\{0,1\}$. We are interested in the number of a certain connected subgraph $S$ of size $k$ in $G$. For ...
2
votes
0answers
126 views

Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?

Disclaimer: This might be an SE question, but I'm not quite sure... Thanks in advance! Setup So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
0
votes
1answer
43 views

Existence of rank-1 weight matrix in some type of deep neural network

Problem This is the first time I have posted a question on this site and it may not be suitable for this venue, which is primarily used for research questions in maths. If someone finds it unsuitable,...
5
votes
0answers
324 views

$\text{Determinant}=(\sum \text{Determinant})^2$

Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition $\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$. QUESTION 1. Is this true? $$\det\...
4
votes
0answers
57 views

Perturbation theory compact operator

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$ $\Vert Kx-\lambda x \Vert \le \varepsilon.$ It is well-known ...
1
vote
1answer
90 views

Counting monomials in skew-symmetric+diagonal matrices

This question is motivated by Richard Stanley's answer to this MO question. Let $g(n)$ be the number of distinct monomials in the expansion of the determinant of an $n\times n$ generic "skew-...
11
votes
2answers
438 views

Classification of algebras of finite global dimension via determinants of certain 0-1-matrices

I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background). A Morita-...
2
votes
2answers
208 views

Equal-valued determinants in search of a proof: Part III

Encouraged by David's proof for my earlier MO question, let's consider a similar problem. I can prove the below equality by computing each of the two sides, directly. That means, there is an ...
11
votes
1answer
392 views

Catalan determinants in search of a proof: Part II

This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof. ...
6
votes
2answers
207 views

Is Gram-Schmidt on a separable Hilbert space operator norm continuous?

Let $\mathcal H$ be a separable Hilbert space, with inner product $\langle\cdot,\cdot\rangle$, and with orthonormal basis $(e_i)_{i\in\mathbb N}$. Consider a continuous linear embedding $A\colon\...
5
votes
1answer
278 views

Determining the primitive order of a binary matrix

Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows $$ {\bf A}_n=\left( \begin{array}{c} 0&0&\cdots&0&0&0&0&1&1\\ 0&0&\cdots&0&0&...
0
votes
0answers
24 views

Determinant of the Sum of a Left Circulant and a nonsingular matrix over field of order 2

For any matrix $A=(a_{ij})\in M_n(\mathbb{F}_2)$, denote $\bar{A}=(\bar{a_{ij}}),$ where $\bar{a}=0$ if $a=1$ and $\bar{a}=1$ if $a=0$. Suppose $n$ is a power of two. Let $A$ (left circulant) and $I'$ ...
5
votes
1answer
107 views

eigenvalues of a symmetric matrix

I have a special $N\times N$ matrix with the following form. It is symmetric and zero row (and column) sums. $$K=\begin{bmatrix} k_{11} & -1 & \frac{-1}{2} & \frac{-1}{3} & \frac{-1}{4}...
7
votes
1answer
125 views

Cycle types of permutations from affine group

Let $V$ be a vector space of dimension $n$ over the field $F=\mathrm{GF}(2)$. We identify $V$ with the set of columns of length $n$ over $F$. Let $G = \mathrm{AGL}(V)$ be a group of affine ...