# Questions tagged [linear-algebra]

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

3,709 questions

**2**

votes

**2**answers

34 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

21 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

14 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

51 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

**0**answers

45 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

**0**answers

56 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

**0**answers

107 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 ...

**6**

votes

**2**answers

332 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

**0**answers

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

**0**answers

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

**0**answers

31 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

**0**answers

167 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

**0**answers

148 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

114 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

**0**answers

42 views

### How to infer information about order of matrix [on hold]

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 ...