Questions tagged [linear-algebra]

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

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-2
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0answers
21 views

finding area of intersection of a rectangle and circle [closed]

I have a rectangle intersecting with a circle. I want to find the area of intersection shown in blue in the figure. I am calculating the area of rectangle as: A = b x (50-d) would that be the ...
3
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1answer
67 views

Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not. \begin{align} (\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1} \end{align} where $A, B$ are Hermitian ...
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0answers
46 views

Image of Frobenius element under irreducible representation is diagonalizable

Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
7
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2answers
108 views

Reference request: continuity of Cholesky factor

It most books dealing with Cholesky decomposition, or it is variants, one finds a statement of the form if $A$ is symmetric $k\times k$ positive semi-definite (non-negative definite) then the $k\times ...
2
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1answer
140 views

The operator equation $AB = \lambda BA$ for self-adjoint operators

Suppose that $A$ and $B$ are self-adjoint bounded linear operators on a Hilbert space and $\lambda \in \mathbb{C}$. It turns out that if $\lambda \notin \{-1, 1\}$ then $AB=\lambda BA \implies AB = ...
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0answers
35 views

Product of matrices has real eigenvalues?

Let $A$ be a (symmetric) positive definite matrix and $\hat{n}$ be an arbitrary unit vector. Consider $b,c,d,k$ arbitrary positive integers. I would like to know if the following matrix has real ...
4
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0answers
51 views

Characterization of “PSD-Squared” Matrices

$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...
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23 views

Is the SVD optimal (by Eckart Young theorem) because maximal data variance is captured from both the column and row space?

I am asking this question seeking validation of an intuitive understanding of the veracity of the Eckart-Young theorem which struck me in my study of the SVD and Principle Component Analysis. The ...
0
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0answers
127 views

Boundedness of total current in electrical network (Banded graph)

Consider the following symmetric matrix (adjacency matrix): $$A=(a_{ij})_{1\leq i,j\leq n}$$ such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
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43 views

Relation between subspaces of diagonal matrix and its “sign” matrix

Let $D$ be a $n \times n$ diagonal matrix with both positive and negative (but all non-zero) entries. Let $J = sign(D)$ be the matrix of $1$s and $-1$s representing the signs of the entries of $D$. ...
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0answers
21 views

Express the length of a projection in terms of the dot product [closed]

I have this question when reading Linear Algebra http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_alg_Lang.pdf, page 39 - 40. Consider a plane defined by the equation (X - P)· N == 0, and ...
9
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2answers
249 views

Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph: $$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$ with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
3
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1answer
62 views

Observable nearly commuting with a “complete” set of commuting observables

Consider the Hilbert space $H = E^{\otimes n}$ where $E=\mathbb{C}^2$. On $E$ we have an observable $O$ (i.e. a Hermitian matrix) that is diagonalizable in the standard basis with eigenvalues $1$ and ...
1
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0answers
41 views

On smooth extensions of functions

Let $f(x) = \left(I - \hat{n}\hat{n}^T \cdot\textbf{1}_{\vec{n}^TAx \geq 0}\right)Ax$, where $I$ is the identity matrix, $A$ is a (symmetric) $d\times d$ positive definite matrix, $\hat{n}$ is an ...
5
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2answers
197 views

Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix

Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...
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0answers
63 views

Computational complexity in linear solvers

I have recently been trying out methods of coding for solving systems of linear equations on Python. Of course, I first used the inbuilt function $\mathit{inv}$ under certain if-conditions to obtain ...
1
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1answer
102 views

Integrality certification for product of two matrices $A B^{-1}$

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...
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0answers
37 views

“euler form” of a unitary matrix? [closed]

A little background: I'm a student of physics (forgive me for a difference in language or the lack of familiarity with this topic), trying to solve an equation of the form: \begin{equation} \mathcal{...
0
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0answers
66 views

Classical papers in linear algebra suitable for an undergraduate reading group?

I'm interested in collecting some of the classical papers in linear algebra over the years. To be more specific, I'm looking for interesting and useful results that extend beyond what is typically ...
0
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0answers
37 views

trace expressions for matrix quadratic forms [migrated]

Let $A$ be a real symmetric $n \times n$ matrix. Which quadratic forms in $A$ can be written in trace form? Such an expression would naturally generalise some invariant random matrix ensembles. ...
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0answers
31 views

Problems with rank of a matrix [closed]

I have some doubts about the rank of this matrix:$ \begin{bmatrix}\cos(x) & 0 & \frac{\sin(x)}{\cos(y)^2} \\ \sin(x) & 0 & \frac{\sin(x)}{\cos(y)^2} \\ \text{tg}(y) & \frac{1}{\cos(...
1
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1answer
94 views

Symmetric tensor components

EDIT: I thought on rephrasing the question in another way: I have been working recently with a tensor that satisfies $A_{ijkl}=A_{i+b,j+b,k+b,l+b}$ $\forall$ $i,j,k,l$ $\in$ Z $$dist(i,j,k,l)\...
1
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0answers
88 views

Phase angles of a complex eigenvector

I have the following system for $\lambda \in \Bbb C, \lambda \neq 0$ and $\pmb{p},\pmb{q} \in \Bbb C^n$, $(\pmb{p}^T, \pmb{q}^T)\neq0$: $$\begin{cases} F(\lambda) \pmb{p} - g(\lambda) \pmb{q} - \...
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0answers
93 views
+50

Question on “semi-linear” dynamical systems

I am interested in understanding the convergence properties of a dynamical system at zero. We know that a dynamical system of the form $x_{k+1} = Ax_{k}$ where $A$ is a symmetric positive definite ...
3
votes
1answer
95 views

Bounds on $\operatorname{sgn}(Au) - \operatorname{sgn}(Av)$ when $\|u-v\|_1 \leq \epsilon$

$\DeclareMathOperator\sgn{sgn}$Suppose A is a $N \times N$ Hermitian and unitary matrix, i.e., $A^{\dagger}=A$ and $A^{\dagger}A=I =AA^{\dagger}$. (Assume all entries are real.) And let $u \in \{-1,1\...
12
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0answers
261 views

Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field

Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
-1
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0answers
81 views

Question on eigenvectors [closed]

Let $\hat{n}$ be a unit vector and $\Delta$ a diagonal matrix of size $m\times m$ with positive entries (at least one entry is greater than one). I would like to show that the sequence of matrices $(I ...
0
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0answers
14 views

Proof for strict separation of the eigenvalues ​of a Jacobian matrix with its minors

Let's consider a jacobi matrix (or tridiagonal symetric matrix where adjacent diagonals coefficients are strictly positive) : \begin{equation} T_n = \begin{bmatrix} a_1 & b_1 & 0 & \...
1
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2answers
229 views

Linear independence of exponential functions: a reference

Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
1
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0answers
57 views

Computational complexity of computing the trace of a matrix product under some structure

I have two problems related to computing some trace, and some (possibly suboptimal) answers. My question is about a potential more efficient algorithm for each one. (More interested in an answer to ...
6
votes
2answers
222 views

Characteristic polynomial of checker matrix

For every integer $n > 0$, let $C_n$ be the $4n \times 4n$ matrix having $1$'s in all positions $(i, j)$ such that $i - j$ is even, $3$'s in the two diagonals determined by $|i - j| = 2n + 1$, and $...
0
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4answers
223 views

Solution for $Xa + X^Tb = c$ where $X^TX = I$?

There are three known $n\times1$ vectors: $a, b, c$, along with one unknown $n\times n$ matrix: $X$. I am only interested in the $n={2,3}$ cases. $X$ is $2\times 2$ or $3\times 3$ rotation matrix ...
6
votes
1answer
79 views

Is the matrix positive definite given the Gauss-Seidel method converges?

I know that the Gauss-Seidel method converges given that the matrix you want to solve is symmetric positive definite. However, I'm wondering if the "converse" of the statement is true. Namely, if $A$ ...
4
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0answers
214 views

Inequalities for trace/eigenvalues of product of multiple 2x2 matrices

Consider the matrix product $\prod_i^n A_i$, where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
4
votes
1answer
154 views

Neighborhood of an orthogonal matrix

Let $A\in O(n)$ be an orthogonal matrix and let $\vec{a}_1,\dots,\vec{a}_n$ be its rows. For a vector $\vec{v}=[v_1,\dots,v_n]$, let $\max(\vec{v})=\max\{|v_1|,\dots,|v_n|\}$. Prove or disprove that ...
1
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0answers
41 views

A real system of bilinear equations with $2n$ unknown and equations

I have the following system of $2n$ bilinear equations, for a square invertible matrix $A \in \mathbb{R}_{n \times n}$, and $2n$ unknowns organized in vectors $x,y \in \mathbb{R}^n$: $$ diag(y) A x = ...
0
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0answers
32 views

pseudo-inverse of short fat matrix

Consider the matrix $A \in \mathbb{R}^{m \times n}$ with $m < n$. Is its pseudo-inverse $A^\dagger = (A^\top A)^{-1} A^\top$ computable ? I'd expect not, because $A^\top A$ is not of full rank ( $...
0
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1answer
57 views

Similar to inverse plus rank 1

Given a real, invertible matrix $A$. For which vectors $b$ and $c$ is $$ A^{-1} + bc^T $$ similar to $A$? And is the rank-1 matrix $bc^T$ unique?
4
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1answer
94 views

Rank of a block of an invertible matrix

Let $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$ be an invertible matrix where $A_{11}$ is square. Let $A^{-1} =: B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} &...
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0answers
57 views

Can we use modified Cholesky's decomposition for solving linear equations?

So I was learning about modified Cholesky's decomposition, specifically Gill, Murray, and Wright algorithm. And although I was able to decompose a negative definite matrix into $LDL^T$, when I used ...
3
votes
1answer
146 views

An orthogonal matrix that satisfies a property must be a permutation matrix

Let $A$ be an $n\times n$ orthogonal matrix such that $\sum_{k=1}^na_{ik}^3a_{jk}=\sum_{k=1}^na_{jk}^3a_{ik}$ for every $1\le i,j\le n$. Original question which is solved by a counterexample given (...
1
vote
1answer
42 views

How can I find minimum and maximum eigenvalue of non-positive define matrix [closed]

There is a power iteration method, but it only returns the greatest(in absolute value) eigenvalue of matrix. So when we have negative eigenvalues it'll give wrong results. Is there any method, which ...
4
votes
0answers
109 views

Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$. I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
0
votes
1answer
37 views

Lipschitz continuity of multivariable function in expected value

Suppose $h:\mathcal{X \times Y \times W}\rightarrow\mathcal{X}$, with $\mathcal{X,Y,W} \subseteq \mathbb R^d$, $d \in \mathbb N$ is Lipschitz in $x,y$, i.e., $$ \| h(x,y,w) - h(x',y',w) \|_2 \le L_h (...
1
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0answers
41 views

Diagonally similar to submatrix of orthogonal matrix

Given $A \in \mathbb{R}^{n \times n}$, with $0 < |\det(A)| < 1$. Does a diagonal matrix $D$ exist such that $$ B = D^{-1} A D $$ is the principal submatrix of an $(n+1)\times(n+1)$ orthogonal ...
0
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0answers
66 views

How to solve a non-local self-consistent equation

I have been struggling lately with solving numerically an equation of the form: $$ g(x\pm x_{0}) = F[ g(x) ] $$ where $g(x)$ is a matrix satisfying the condition $g(x\to\pm\infty)=0$. My question is ...
1
vote
0answers
35 views

Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices

I have the following problem: I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$. The first is a regular Toeplitz matrix $A$...
2
votes
1answer
142 views

How to find eigenvalues following Axler?

Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple: Fix ...
1
vote
1answer
62 views

$\arg\max$ in the dual norm of the nuclear norm

Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by $$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$ whereas the nuclear norm is ...
0
votes
0answers
60 views

Probability that random points are affine subspace

I asked this question in Math stack exchange, but I think it is more relevant here. Suppose that $\mathbb{F}_q$ is a finite field with $q$ elements. Let $U = \{u_1, \ldots,u_m\}$ be a set of $m$ ...

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