Skip to main content

Questions tagged [skew-symmetric-matrices]

Matrix $A$ is skew-symmetric (or antisymmetric) iff $A^\top = -A$.

Filter by
Sorted by
Tagged with
-1 votes
0 answers
17 views

Anti-intuition kurtosis

I want to describe the tailedness of distribution and calculated the kurtosis of "before" and "after", and it seems like "after" has a larger kurtosis than "before&...
lisunet's user avatar
1 vote
1 answer
92 views

A general solution $M$ for the matrix equation $A^* M^* + M A = 0$

As titled, for some unknown complex-valued matrix $A$, possibly rectangular, does there in general exist a complex-valued matrix $M$ such that $$ A^* M^* + M A = 0 $$ Here, $A^*$ denotes the ...
user594147's user avatar
1 vote
1 answer
40 views

Skew-symmetric matrix least-square problem.

Is there a simple solution to the following problem: Given $x,y\in\mathbb{R}^n$, $$ \min_{A\in M^n(\mathbb{R})} \left\| \left(\frac{A-A^t}{2}\right)x - y\right\|^2 $$ Or equivalently $\min ||Ax-y||^2$...
Baptiste GENEST's user avatar
0 votes
1 answer
138 views

$\det(A^2-B^2) \leq \det(A^2)$ when $B$ is of rank 1

This is an exercise problem given at linear algebra class. As a problem before this I was able to show that $$\det(A + uv^T) = \det A + v^T (adj A )u$$ when $A$ is order $n$ square matrix and $u,v$ ...
mathhello's user avatar
  • 918
0 votes
0 answers
44 views

Show that $A\in M_n(\mathbb R)$ is skew-symmetric if $\det(A+M)=0$ for all M skew-symmetric, with $n$ odd.

Let $n$ be an odd integer, and $A\in M_n(\mathbb R)$ such that, for all $M\in M_n(\mathbb R)$ skew-symmetric, we have: $$ \det(A+M)=0$$ Show that $A$ is skew-symmetric. What I did so far: Nothing much ...
fus3r's user avatar
  • 173
3 votes
3 answers
196 views

The relation between the eigenvalue of a Hermitian matrix and the block matrix that composed by it real and imaginary part

Recently I am reading a paper. In their "Proof of Lemma 1" on page 24, they have: $$\lambda_+(\mathbf{Q})=2\lambda_+(\tilde{\mathbf{Q}})$$ where $\mathbf{Q}$ is a Hermtian matrix, $\tilde{\...
tyrela's user avatar
  • 353
0 votes
0 answers
56 views

Zero trace matrix, product of symetric and antisymetric matrices

It is well-known that every square matrix can be written as the sum of a symetric matrix and an antisymetrix matrix. The same does not hold for the product : for example, matrices $M\in\mathscr{M}_n(\...
P.Fazioli's user avatar
  • 243
0 votes
1 answer
100 views

Can the product of a complex symmetric unitary matrix and a skew-hermitian matrix be complex skew-symmetric?

Let $S$ be a nonzero complex symmetric unitary matrix ($S\neq 0$, $S = S^T$, $S^HS=Id$) Let $B$ be a nonzero skew-hermitian matrix ($B\neq 0$, $B^H = -B$) Can their product $P\triangleq SB$ be ...
LucLeMag's user avatar
0 votes
3 answers
79 views

Skew-symmetric matrices and the fact that their exponential matrix is orthogonal

I want to ask why "If $A$ is skew-symmetric ($A^T=-A$) then $e^{At}$ is an orthogonal matrix". Here is my solution step: $e^{At}*(e^{At})^T =e^{At}*e^{-At}=e^{At-At}=e^0$ I think the matrix $...
Ruizhe Pang's user avatar
1 vote
1 answer
42 views

Manipulating skew-symmetric matrices

Consider the following object $$M_{ij} = A_{il}B_{lj}-A_{jl}B_{li}$$ where repeated indicies imply summation and $A$ and $B$ are invertible anti-symmetric matrices. Now, clearly, $M$ is anti-symmetric ...
Dr. user44690's user avatar
0 votes
0 answers
33 views

Help in understanding why the antisymmetric part of the positive semi-definite quadratic form does not contribute?

Can you help me understand the statement given below, particularly what is said when evaluating (15) and the conclusion leading to, and given by, (16)? On a mathematical level, in view of the scalar ...
Armadillo's user avatar
  • 535
0 votes
0 answers
55 views

Is the product $B D B^T$ always a symmetric tridiagonal matrix? Where $D$ is a diagonal matrix and $B$ a sparse matrix.

I have a diagonal matrix ${\bf D}_{n \times n}$ and a rectangular matrix ${\bf B}_{m \times n}$ where $n \gg m$. All but $m$ rows of ${\bf B}$ have non-zero elements. These $m$ rows have only six non-...
Olumide's user avatar
  • 1,251
-1 votes
1 answer
89 views

If $A$ is a skew symmetric matrix, is it possible to always find a vector $x$ such that $x - x^\top = A$

If $A \in \mathbb{R}^{d \times d}$ is a skew symmetric matrix, is it possible to always find a vector $x \in \mathbb{R}^d$ such that $x1^\top - 1x^\top = A$, where $1$ is a vector of all ones $\in \...
Joff's user avatar
  • 916
1 vote
1 answer
71 views

Proof a Property About Pfaffian Acting on Block Diagonal Matrices

We have been told in class that the Pfaffian of $2n$ by $2n$ skew-symmetric matrix $A$ is defined as: $$Pf(A)=\frac{1}{2^nn!}\sum_{\sigma\in S_{2n}}A_{\sigma(1)\sigma(2)}\cdots A_{\sigma(2n-1)\sigma(...
SuperSupao's user avatar
0 votes
0 answers
40 views

Sort Skew-Symmetric Tridiagonal Matrix

Suppose I have a skew-symmetric tridiagonal matrix of the from \begin{equation} M = \begin{pmatrix}0 & \lambda_1 & 0 & 0 & 0 &\cdots\\ -\lambda_1 & 0 &\...
Kestrel's user avatar
  • 31
0 votes
1 answer
70 views

Property of skew symmetric matrix [closed]

Prove that the determinent of a skew symmetric matrix of even order remains unchanged if the same number is added to all the elements
Prathamesh's user avatar
-1 votes
1 answer
99 views

How many groups with $100$ elements exist, where for min. $80\%$ of the pairs $(a, b)$ is $ab = cba$, for some constant $c$? [closed]

Got the idea from skew symmetry of matrice rings, where when you swap 2 elements you need to add a minus sign. But I was wondering to what extend is this possible for only a finite group. I guess I ...
Skewbinger's user avatar
2 votes
2 answers
110 views

Find a matrix representation of $(fL)(x,y) = L(Ax, Ay)$, where A $\in \mathbb{R}^{3 \times 3}$

I'm interested in the following question: Let $L((\mathbb{R}^3)^2; \mathbb{R})$ be the vector space of bilinear maps from $\mathbb{R^3}\times \mathbb{R^3}$ into $\mathbb{R}$. Consider the subspace V $...
Anonymous73648's user avatar
6 votes
1 answer
357 views

Let $A$ be a skew-symmetric real matrix, prove that there exists a vector $x\ge0$ such that $Ax\ge0$ and $Ax + x > 0$

This is an assignment that I am struggling with. Let $A$ be a skew-symmetric real matrix, prove that there exists a vector $x\ge0$ such that $Ax\ge0$ and $Ax + x > 0$. Not sure how to proceed ...
Aster's user avatar
  • 61
0 votes
0 answers
90 views

How to block-diagonalize a skew symmetric matrix

I have encountered the fact on wikipedia that every skew-symmetric matrix can be block-diagonalized, where the matrix is in the form indicated by the following picture. I was wondering if there is a ...
Tony Deng's user avatar
0 votes
2 answers
123 views

Product of a positive diagonal matrix with a skew symmetric matrix yields a matrix with imaginary eigenvalues.

For a skew symmetric matrix $A=-A^T$, and a diagonal matrix $D=diag(d_{ii})$ such that $d_{ii}\in (0,\,1]$. Since $A$ has imaginary eigenvalues or a zero eigenvalue, will $DA$ also have imaginary ...
Desperado's user avatar
0 votes
0 answers
63 views

Skew-symmetric matrix absolute value and row/column sums

I'm looking at an algorithm that takes a square matrix $M$ containing real numbers and calculates the skew-symmetric part like so: $ S = (M - M^t)/2 $ It then calculates the absolute value of $S$: $ ...
anjama's user avatar
  • 153
2 votes
1 answer
309 views

Eigenvalues and eigenvectors of skew-symmetric matrix

Let $A$ be the $N\times N$ skew-symmetric matrix defined by the rules $$\begin{cases}A_{ij}=1 & \text{when }i<j, \\ A_{ii}=0 & \forall i=1,...,N \\ A_{ij}=-1 &\text{when }i>j.\end{...
Ghost of Ludwig Boltzmann's user avatar
1 vote
1 answer
70 views

For skew-symmetric $U$, prove that $U(1-U^\dagger U)^{-1}$ is, too

Let $U$ be a complex skew-symmetric matrix ($U = -U^\mathsf{T}$, where $U^\mathsf{T}$ means the transpose of $U$). I want to show that $U(1 - U^\dagger U)^{-1}$ is also skew-symmetric. Here $U^\dagger$...
Zhengyuan Yue's user avatar
0 votes
0 answers
37 views

Definition of the (anti-)symmetric part of a tensor

Consider the following tensor: $$ A_{ij} = B_{k,i}C_{k,j} $$ which can be decomposed into its symmetric and antisymmetric parts: $A_{ij} = A_{ij}^s + A_{ij}^a$. I'm all confused as to how to express ...
Bjaam's user avatar
  • 77
0 votes
2 answers
61 views

Eigenvalues of skew-symmetric matrices from 2D random points

I am generating $n$ random points in two dimensions $(x_i, y_i)$. Then I form this skew-symmetric matrix $$ M_{ij} = \begin{cases} x_i y_j - x_j y_i & \text{if } i < j \\ 0 & \text{if } i=...
user2167741's user avatar
1 vote
0 answers
43 views

Is there an example that a specific matrix has Jordan block $J_{2}(i)$?

Consider a matrix of the form $\begin{bmatrix} A & C\\ -C^{T} & B \end{bmatrix}$ where A and B are symmetric matrices. Can matrices of this type have a Jordan normal form representation ...
Kyuwon Kim's user avatar
2 votes
2 answers
104 views

Can any symmetric matrix with zero diagonals be written as the commutator of a skew-symmetric and diagonal matrix with positive diagonals?

$\newcommand{\S}{\mathcal{S}}$ Question: Let $A$ be a symmetric matrix with zero diagonals. I was curious if there exist a diagonal matrix with positive diagonals $D$ and a skew-symmetric matrix $\...
Spencer Kraisler's user avatar
0 votes
1 answer
432 views

Show that the determinant of a 4x4 skew-symmetric matrix is non-negative [closed]

I've tried expanding along rows and columns with general variables which only brought a mess and didn't get me anywhere. The mess decreased a little when I used the fact that the determinant of an odd ...
struggling_undergrad's user avatar
2 votes
1 answer
316 views

If $A$ is $n \times n$ skew-symmetric matrix where $n$ is an odd integer number, then $A$ is singular.

In order to answer, I must first state whether the statement is true or false, followed by an explanation If $A$ is $n \times n$ skew-symmetric matrix where $n$ is an odd integer number, then $A$ is ...
xlu_uk dev's user avatar
1 vote
1 answer
65 views

Why is the transport operator skew-symmetric?

In the context of this paper, where $$ T:=v\cdot\nabla_x-\nabla_xV\cdot\nabla_v $$ is a transport operator, it is said on page 3, that $T$ is skew-symmetric with respect to $\langle\cdot,\cdot\rangle$,...
selector's user avatar
  • 447
1 vote
1 answer
174 views

Antisymmetric polynomials in two variables

In Prove that every symmetric polynomial can be written in terms of the elementary symmetric polynomials it is argued that a symmetric polynomial of two variables $x,y$ can be written as a sum over ...
Rubilax96's user avatar
2 votes
2 answers
153 views

Prove: if $\bf{AB^T}$ is skew-symmetric and $\bf A$ full-rank, then $\bf{AX}=\bf B$ has unique solution $\bf X$

I've run into this statement while trying to prove that the energy of a rotating body in $N$ dimensions is conserved, it's the last puzzle piece I'm missing. Let $\bf A$ and $\bf B$ be two $M \times N$...
Gabi's user avatar
  • 215
1 vote
1 answer
70 views

Is it always true that matrix representation of a skew-symmetric non-degenerate bilinear form is orthogonal with respect to some basis?

I am going through the definition of symplectic matrices. While searching through various properties of those class of matrices I found that every symplectic matrix is necessarily invertible and the ...
Anacardium's user avatar
  • 2,502
0 votes
2 answers
124 views

Manipulation of a multiple cross product expression using skew symmetric matrices

In the derivation of equation (8.23) in the book "Modern Robotics: Mechanics, Planning, and Control" there are some manipulations of the term for the moment contribution of the centripetal ...
Greg Jones's user avatar
0 votes
3 answers
260 views

Optimization Problem with a skew-symmetric matrix as a variable

I'm currently trying to solve the following optimization problem $$\mathop{\text{minimize }}_{X \in \mathbb{R}^{n \times n}} \left\| A X - B \right\|_F^2 \quad \text{ subject to } X = -X^T$$ in which ...
FEDERICO BENZI's user avatar
3 votes
4 answers
205 views

Intuition for why any square matrix is a sum of a symmetric matrix and a skew-symmetric matrix? [closed]

I've seen proofs for this theorem and understood then. However, I feel like I can gain a better intuition. I would never think of this theorem myself. How could someone discover this theorem? Is ...
Gal A.'s user avatar
  • 147
0 votes
1 answer
52 views

Prove that in case of a skew-symmetric matrix, $A$, we have, $A^t=-A$.

I cam accross the following definition in Wikipedia https://en.m.wikipedia.org/wiki/Skew-symmetric_matrix , which is attached below , for a reference: However, I dont get how the two definitions i.e ...
Arthur's user avatar
  • 2,614
0 votes
0 answers
39 views

Amount and Indices of Skew Diagonals in n-dimensional structures

Suppose you have an $n ^ m$ matrix, in this example $3 ^ 2$. Each of those will have a $1$ at the "start", then increase linearly. $\left( \begin{array}{rrr} 1 & 2 & 3 \\ 2 & 3 &...
Thomas B.'s user avatar
  • 187
1 vote
1 answer
93 views

Relationship between the spectrum of a skew-symmetric matrix and its symmetric counterpart

Suppose $A$ is a real skew-symmetric matrix, and $\tilde{A}$ is an induced symmetric matrix created by flipping the sign of all elements below the diagonal. As we know, the spectrum of $A$ is purely ...
nalzok's user avatar
  • 804
1 vote
0 answers
57 views

How does 2a = 0 not imply a = 0 for a random field affect the fact that the set of n×n symmetric and skew matrices are a subspace and their dimension?

For a homework assignment I'm supposed to prove that the set of n×n symmetric matrices and the set of n×n skew matrices with the elements of the matrix being elements of a random field F are a ...
Fregheit Meier's user avatar
1 vote
1 answer
90 views

If the matrix exponential is unitary, is the exponent necessarily skew-Hermitian?

If $A$ is skew-Hermitian, then $e^A$ is unitary. But is the converse true? That is, if $e^A$ is unitary, is $A$ necessarily skew-Hermitian?
ashpool's user avatar
  • 6,966
2 votes
1 answer
183 views

If $A$ and $B$ are skew-symmetric, then $A^2BA$ is symmetric

Let $A$ and $B$ be skew-symmetric $3 \times 3$ matrices. That is, $A^t = -A$ and $B^t = -B$. According to WolframAlpha, $A^2BA$ is a symmetric matrix. I believe there is an easier way to prove this ...
D.L's user avatar
  • 452
0 votes
0 answers
70 views

System of first order differential equation objective question.

Let $A\in M_3(\mathbb R)$ be a skew-symmetric matrix and $x:[0,\infty)\to\mathbb R^3$ be a solution of $$x’(t)=Ax(t),\forall t\in (0,\infty)$$ Which of the following statements are true? $1.$ $\|x(t)\|...
neelkanth's user avatar
  • 6,058
1 vote
1 answer
51 views

A skew symmetric and orthogonal matrix has eigen values (3/5) + (4i/5). How can this be possible? It must have 0 or purely imaginary values. Problem 1

Problem 1. It is Orthogonal and skew symmetric but eigen values aren't purely imaginary or zero Are the following matrices symmetric, skew-symmetric and/or orthogonal? $$\frac15\begin{bmatrix}3&-4\...
aditya pawar's user avatar
1 vote
0 answers
100 views

Show that $A + A^{T} = 0$ iff $x^{T}Ax = 0$. [duplicate]

In the first direction, I know that $A + A^{T} = 0$ implies that all diagonal entries must be zero and each non-diagonal has a negative entry, therefore the entire product $x^{T}Ax = 0$ must be equal ...
No_Bass's user avatar
  • 67
4 votes
1 answer
119 views

Question regarding determinants of a sequence of anti-symmetric matrices.

I have a sequence of real anti-symmetric matrices $M(k),k=1,2,\dots$, where $M(k)$ is a $2k\times 2k$ matrix with $(i,j)$ th element defined as $$ M(k)_{ij} =\frac{i-j}{i+j},\,1\le i\le2k,\, \,1\le j\...
user10001's user avatar
  • 463
2 votes
0 answers
136 views

Pfaffian definition for a complex skew-Hermitian matrix?

Normally, the Pfaffian is defined for real skew-symmetric matrices $A = -A^T$, and some authors extend this to complex skew-symmetric matrices. Is there a straightforward generalization of the ...
SGJ's user avatar
  • 377
0 votes
1 answer
56 views

Prove that the third partial derivative over the space is a formally skew-adjoint operator in $\mathbb{R}$

I'm trying to solve the following question: Prove that $\mathcal{J} = \frac{\partial^3}{\partial z^3}$ is formally skew-adjoint on the space $C^ \infty(I, \mathbb{R}) $ of smooth real-valued ...
olenscki's user avatar
  • 125
0 votes
0 answers
35 views

Commutator of Skew-Symmetric Matrices with Integer entries

Let $\mathcal{S_3}$ be the collection of $3\times 3$ skew-symmetric matrices with integer entries. There is no non-zero $D\in \mathcal{S_3}$ that satisfies the following equation $$[D,[D,[D,N]]]=0$$ ...
Elliot's user avatar
  • 1

1
2 3 4 5