# Questions tagged [linear-algebra]

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

5,637
questions

4
votes

0
answers

102
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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 ...

4
votes

1
answer

71
views

### 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}
\...

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 (...

2
votes

1
answer

55
views

### 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 \...

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 ...

0
votes

1
answer

151
views

### 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 $\...

-2
votes

0
answers

76
views

### 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, ...

1
vote

1
answer

94
views

### 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, \...

0
votes

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. ...

0
votes

1
answer

101
views

### “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 ...

-4
votes

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 $A(x,y)$ ...

2
votes

1
answer

61
views

### 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 ...

0
votes

0
answers

62
views

### 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 ...

0
votes

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 ...

3
votes

1
answer

85
views

### 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 ...

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 $...

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 ...

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 ...

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,...

-1
votes

0
answers

67
views

### 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 ...

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 ...

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)-\...

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: ...

0
votes

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\...

9
votes

0
answers

2k
views

### 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 $...

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 ...

0
votes

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 $1\leq i<j\leq d$ ...

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}{ \...

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 ...

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 \\
...

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, ...

0
votes

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 ...

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 ...

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 ...

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}^...

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 ...

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 ...

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 ...

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 ...

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 ...

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 - \...

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 ...

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 $...

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 ...

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 $||Bx||_\infty \leq ||x||_\infty$....

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://...

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:...

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 ...

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 ...

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},
...