# Questions tagged [linear-algebra]

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

3,709 questions

**2**

votes

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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

**0**answers

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

**0**answers

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

**1**answer

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 ...

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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 ...

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vote

**0**answers

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

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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

**2**answers

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 ...

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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 ...

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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 ...

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**0**answers

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**

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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**

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**0**answers

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

**1**answer

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 $\...

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**0**answers

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

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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

**1**answer

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\...

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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

**0**answers

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 ...

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votes

**0**answers

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)& -\...

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votes

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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

**1**answer

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

**3**answers

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

**2**answers

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

**1**answer

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

**0**answers

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 ...

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votes

**0**answers

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-...

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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 ...

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votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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 $V^*\otimes W^*$ ...

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vote

**0**answers

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\...

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votes

**1**answer

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

**1**answer

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 ...

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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

**1**answer

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,...

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**0**answers

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\...

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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 ...

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vote

**1**answer

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

**2**answers

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-...

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votes

**2**answers

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

**1**answer

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.
...

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votes

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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 ...