Questions tagged [linear-algebra]

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

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Continuity of eigenvector of zero eigenvalue

Wonder whether anyone has an idea on showing the following or to point out that it is not true: Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
muddy's user avatar
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What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?

Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation \begin{align*} & X = A X A^T + \operatorname{Id} \tag{1} \...
Tardis's user avatar
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2 votes
0 answers
37 views

Perron-Frobenius theory for operators on matrices

Let $A$ be a Hermitian linear operator on the space of $n\times n$ complex matrices. Let's suppose $A$ is "non-negative" in the sense that it preserves the cone of non-negative definite (...
AdamNie's user avatar
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2 votes
1 answer
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Symmetric and anti-symmetric matrices and maximal eigenvalues

Suppose we start with a symmetric $n \times n$ matrix $A$, the elements of which are either $1$ or $0$. All the diagonal elements of this matrix are set to be $0$. So, $\lambda_{\text{max}}=\sup \...
Alapan Das's user avatar
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9 votes
3 answers
250 views

$G$-module structure of the relation module for a presentation of a finite group $G$

Let $F_n$ be a free group of rank $n\ge 2$, and $F_n\rightarrow G$ a surjection with $G$ finite. Let $R$ be the kernel. From this, we get an action of $G$ on the abelianization $R/R'$ (a free abelian ...
stupid_question_bot's user avatar
0 votes
1 answer
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Reflections on subspaces of $\text{codim} > 1$

Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$. Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
Bipolar Minds's user avatar
-2 votes
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Linear Independency [closed]

Wonder whether the following question is suitable for this forum: Suppose $x_i(t)$, $i = 1, \ldots, n$, are $n$-differentiable over an interval, and they are linearly independent over the interval, ...
muddy's user avatar
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1 vote
1 answer
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Does a random matrix over $\mathbb{Z}_q$ map linearly independent vectors to statistically independent vectors?

Suppose $A \in \mathbb{Z}_q^{n \times m}$ is a random $n \times m$ matrix whose entries are i.i.d. uniform over $\mathbb{Z}_q=\mathbb{Z}/q\mathbb{Z}$, where $q\geq2$. Let $\mathbf{x}_1, \ldots, \...
aleph's user avatar
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1 answer
129 views

Show that $\mathrm{PSL}_2(C)$ is complex algebraic [closed]

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\im{im}$I meet this problem when reading Artin's book Algebra. ...
Math Diego's user avatar
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“Smallest” non-zero linear combination of vectors to obtain a non-negative vector

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix} where $x_{j} \geq 0$. Suppose that ...
Siddharth Iyer's user avatar
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0 answers
97 views

Recognizing a group and another structure [closed]

In some physical modelling I find a mathematical structure of this form: letting $x, y, z \in \mathcal{X}$ with $\mathcal{X}$ a discrete set (in fact, the edges of a graph) I have an object $A(x,y)$ ...
tomate's user avatar
  • 495
2 votes
1 answer
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Lipschitz continuity of eigenprojections

This question has the same flavor of this and this questions, but asks for something stronger. Assume that $A$ is a symmetric $n \times n$ matrix, $H$ is a $n \times n$ perturbation matrix. Moreover ...
Guanaco96's user avatar
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Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
pallab1234's user avatar
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0 answers
22 views

Application of greedy approach for optimization

I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$ where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
Prakirt Raj's user avatar
3 votes
1 answer
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Is there a closed-form solution for $\max_D \operatorname{Tr}(ADBD)$

Is there a closed-form solution for $$\max_D \operatorname{Tr}(ADBD)$$ where $D$ is a $N\times N$ diagonal matrix with $m<N$ number of $1$'s and the rest are $0$'s, and $A$ and $B$ are real ...
CWC's user avatar
  • 351
1 vote
2 answers
120 views

Right inverse of integer matrix

If I have a rectangular matrix $A$ (say $4 \times 6$) with integer entries, is there a way to tell whether it has a right inverse that also has integer entries. I know that if $AA^T$ has determinant $...
user61388's user avatar
5 votes
0 answers
126 views

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products

Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
RandomTensor's user avatar
1 vote
1 answer
158 views

Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant

Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
loup blanc's user avatar
  • 3,564
0 votes
1 answer
105 views

Loss of degree for polynomials

Let $q$ be a power of a prime $p$, $m$ be an integer greater than $0$. Does there exist polynomials $P_0,\cdots,P_m$ of $\mathbb F_q[T]$ not all in $\mathbb F_q$ such that there exist polynomials $Q_0,...
joaopa's user avatar
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-1 votes
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Proposition on Artinian Modules

While I am reading the proof of the following proposition, I don't understand the following: The proposition is: "any non-empty family of submodules of an $R$-module $M$ has a minimal element if ...
Mousa Hamieh's user avatar
2 votes
0 answers
133 views

Spectrum of an almost Hamiltonian matrix

I have a complex-valued block matrix $N=\begin{bmatrix} A & B \\ C & -A^* \end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian. If $C$ were Hermitian, $N$ would ...
mathamphetamine's user avatar
0 votes
1 answer
65 views

Matrix quantization and effect on singular values

Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for $$ \| \sigma_i(A)-\...
ABIM's user avatar
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4 votes
1 answer
223 views

Does a generic linear map admit a vector whose iterates span $V$?

We say a linear map $T$ on a finite dimensional vector space $V$ admits spanning vectors if there exists some vector $v \in V$ whose iterates $v, Tv, T^2 v, \dots$ under $T$ span $V$. Question: ...
Nate River's user avatar
  • 4,604
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0 answers
102 views

Generalization of SVD algorithm

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\...
Martin Clever's user avatar
9 votes
0 answers
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Is it legitimate to use a delta function as a composite function? [migrated]

I am used to $\int f(x) \delta(x) \, dx = f(0) $ and the idea that this represents the action of a dual vector on $f(x)$ as an element of a vector space of functions, mapping it to the real number $...
Robert Lyon's user avatar
0 votes
1 answer
81 views

Find efficiently greatest difference between $2$ vectors from set of vectors [closed]

Let us have a list of vectors in a $3$D space. Is there a more efficient way to find the greatest difference between any two of them than combining each, computing the size of their difference, and ...
Honza S.'s user avatar
  • 109
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0 answers
25 views

The selection of minimal generating sets in Lie algebra

Suppose $A$ is a Lie algebra on field $F_{p}$ with $[A,A,A]=0$. Denote $\{a_{1},\cdots,a_{d}\}$ is a minimal generating set of $A$.It's possible that $[a_{i},a_{j}]=0$ for some $1\leq i<j\leq d$ ...
gdre's user avatar
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2 votes
0 answers
187 views

Is there a geometric or calculus-based reason why the following system of equations should have only one solution?

Let $x_1,x_2,x_3,x_4>0$. Consider the following cyclic system of equations: $$ 2 + x_2 + x_3 + x_4 + x_2 x_3 x_4 - 2 \left( \frac{x_2}{ \sqrt{x_1 x_2}} + \frac{x_3}{ \sqrt{x_1 x_3}} + \frac{x_4}{ \...
matilda's user avatar
  • 90
2 votes
0 answers
67 views

The rank of a certain linear combination of mutually commuting nilpotent matrices

Let $A_1,\ldots,A_r$ be mutually commuting $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers. For any complex number $c$, let $A(c):=A_0+cA_1+c^2A_2+\ldots +c^rA_r$. We ...
sagnik chakraborty's user avatar
7 votes
2 answers
547 views

A curious equation on determinant----linear algebra or algebraic geometry?

I recently find a curious and unexplainable(as seems to me) equation on determinant as follows. $$3\begin{vmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ ...
LichenSDU's user avatar
  • 143
1 vote
1 answer
324 views

Solvability of $A X B=C$ with $X=X^\mathrm{T}$

I am studying symmetric solutions to the complex matrix equation \begin{equation} A X B=C, \end{equation} where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
Juan's user avatar
  • 61
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0 answers
79 views

Totally isotropic space for bilinear pairing over ring

A duplicate of this: Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$ a totally ...
JBuck's user avatar
  • 173
4 votes
0 answers
137 views

Does an instance of this generalisation of the determinant exist?

Let $n$ be composite, $d$ a divisor greater than $1$ and $m=n/d$. Does anybody know if there is a general mapping $T$ from $n×n$ matrices to $m×m$ matrices that preserves the determinant? Over a field ...
Maarten Havinga's user avatar
1 vote
0 answers
78 views

Interpreting positive semidefinite matrix as a graph

Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
wsz_fantasy's user avatar
3 votes
1 answer
313 views

"Totally real" linear transformations

Identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$ via the equality $$(z_1, z_2, \ldots, z_n)=(x_1, \ldots, x_n, y_1, \ldots , y_n)$$ Where $z_j=x_j + iy_j$. We call a linear invertible map $A: \mathbb{R}^...
user avatar
1 vote
0 answers
58 views

A criterion for divisors of degree $n$ on the projective line to belong to a linear system of codimension 1

My question is essentially about linear dependence/independence of polynomials, but I will formulate it in the language of algebraic geometry, hoping someone may suggest a result in algebraic geometry ...
Malkoun's user avatar
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0 votes
0 answers
134 views

Simultaneous triangulation and Jordan normal form of commuting nilpotent matrices

Let $A_1,\ldots,A_r$ be $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers, satisfying $A_i\cdot A_j=A_j\cdot A_i$ for all $i,j$. As the matrices commute, they admit ...
sagnik chakraborty's user avatar
3 votes
1 answer
131 views

Existence of a density

Let $W: \mathbb R^d \to \mathbb R^{d \times d}$ be a matrix-valued function with domain $\mathbb R^d$ and taking values in the set of $d \times d$ real matrices, such that $W(x)$ is positive-definite ...
Aurelien's user avatar
  • 191
2 votes
1 answer
199 views

Is matrix B obtained from matrix A?

Assuming a matrix $\mathbf{A} \in \mathbb{R}^{4096 \times 4096}$ sampled from a standard normal distribution $N(0, 1)$, and another matrix $\mathbf{B} \in \mathbb{R}^{4096 \times 4096}$ either sampled ...
eternity's user avatar
0 votes
0 answers
149 views

Generalization of polynomial coefficients

I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
Cardstdani's user avatar
1 vote
0 answers
27 views

Characterization of Gaussian Gram matrices

From Euclidean geometry we know that a matrix $C$ is a matrix of squared Euclidean distances between some points if and only if $-\frac{1}{2} H D H \succeq 0$ (positive semi-definite) with $H = (I - \...
Titouan Vayer's user avatar
0 votes
1 answer
91 views

Matrix-order derivatives (differentiating a function a matrix number of times)

I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
charlesalexanderlee's user avatar
1 vote
0 answers
79 views

Functional inequality with complex variables

I am interested in considering a function $C(\tau)$ of the complex variable $\tau = t + i\eta$ such that $C(\tau)$ is analytic for $\Re(\tau)=t>t_0\ge0$ $\exists$ a constant $C_0$ and a function $...
knuth's user avatar
  • 29
0 votes
0 answers
78 views

Separating orthogonal vectors in $\mathbb{C}^2$

Is it possible to partition $\mathbb{C}^2$ into two sets $S$ and $S'$ such that, given any two nonzero orthogonal vectors $\mathbf{v}$ and $\mathbf{w}$ of $\mathbb{C}^2$, one of them lies in $S$ and ...
GaussJordan's user avatar
5 votes
1 answer
77 views

Interpolation between two matrices so that $L^p$ norm is controlled

Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
tommy1996q's user avatar
0 votes
1 answer
370 views

What is the mathematician's definition of the determinant? [closed]

I am trying really hard to find a good definition of the determinant. I have looked virtually every single resource online and everybody gives a different answer: sum of cofactors or minors https://...
Olórin's user avatar
  • 179
4 votes
0 answers
83 views

The spectra of Hodge-Laplace operators

If we have a sequence of linear maps and finite dimensional inner product spaces $$X\xrightarrow{f} Y\xrightarrow{g}Z$$ such that $g\circ f=0$, then we can consider the Hodge-Laplace operator $$\Delta:...
Mariano Suárez-Álvarez's user avatar
0 votes
0 answers
17 views

Efficient Solution for tridiagonal solving with repeated coefficient lines

I working to speedup calls to LAPACK dgtsv for a specific case, where the the coefficients lines have 2 blocks of repeated coefficients and 3 distinct lines (first, "border" and last) First ...
Yair Lenga's user avatar
0 votes
0 answers
28 views

How to synthetize a controller $\dot{u} = F x + G u$ which stabilizes $\dot{x} = Ax + Bu$?

$\textbf{Introduction}$: I study linear control theory. Among strategies, we begin with vector field $Ax + Bu$, $A \in M_{n^2}(\mathbb{R})$, $B \in M_{n \times m}(\mathbb{R})$, and synthesize a ...
Usuário 6789's user avatar
0 votes
1 answer
60 views

Can non-periodic discrete auto-correlation be inversed?

I'm trying to understand whether discrete auto-correlation can be reversed. That is, we are given $t_0, \dots, t_n \in \mathbb C$ and a set of equations $$ t_{k} = \sum\limits_{i=0}^{n-k} b_i b_{i+k}, ...
Oleksandr  Kulkov's user avatar

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