Questions tagged [skew-symmetric-matrices]

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

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A problem from research paper on skew-symmetric matrices

Given statement is Note: In this research paper all calculation is under field of characteristic $3$ unless specified. $\pmb{proposition:}$ Let $K$ be a field of arbitrary characteristics. Let $X$ be ...
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Maximum dimension of a subspace consisting only of non-degenerate alternating bilinear forms over finite fields.

Let $V$ be a vector space over a finite field $\mathbb{F}_q$. The following result seems well-known in the literature of bilinear forms. The maximum dimension of a subspace consisting only of non-...
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A question about skew-symmetric matrix.

I have no idea how to prove/disprove this Statement. Statement: Let $K$ be a field of arbitrary characteristics. Let $X$ be a skew-symmetric $(n \times n)-$ matrix over $K$ and $H=(h_{ij})$ be skew-...
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Let $A,B\in M_3(\mathbb{C})$ be invertible matrices such that $AB=BA=X$, $A^{T}+A=B^{T}+B=X^{T}+X$. Then, is $\det(X - I) = 0$?

Let $A,B\in M_3(\mathbb{C})$ be invertible matrices such that $AB=BA=X$, $A^{T}+A=B^{T}+B=X^{T}+X$, then: (A) $A=B$ (B) $\det(A-I)=0$ (C) $\det(B-I)=0$ (D) $\det(X-I)=0$ My working: $AB+(AB)^T=X+X^T\...
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If $[a]_\times$ is a matrix such that $[a]_\times b$ = $a \times b$, and $R$ is a (rotation) Matrix, how to simplify $[Ra]_\times$?

Suppose $a \in \mathbb{R}^3$, $b \in \mathbb{R}^3$ and $R \in \mathbb{R}^{3\times3}$. Defining the operation $[a]_\times = \begin{bmatrix} 0 & - a_{z} & a_{y}\\a_{z} & 0 & - a_{x}\\- ...
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1answer
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Skew Product of Symmetric Matrices

Let $A,B$ be two real $n\times n$ symmetric matrices. Is it true that $AB=-BA$ implies $AB=0$? Note that this condition is equivalent to $AB=-(AB)^T-B^TA^T=-BA$, i.e. it is equivalent to $AB$ being ...
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A question related to skew symmetric matrix and Orthogonal matrices

Consider the following problem asked in a masters exam for which I am self studying. Write V for the space of $3 \times 3$ skew - symmetric real matrices. (A) Show that for $A\in SO_3(\mathbb{R})$ ...
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Using a Schur complement, prove that the matrix has two double eigenvalues

For a skew symmetric block $n \times n$ matrix $B$, prove that matrix $M$ has two double eigenvalues. $$ M = \begin{bmatrix} I & B \\ B & I\end{bmatrix} $$ For a proof, I was using the ...
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If a symmetric matrix commutes with all symmetric matrices, is it then a multiple of the identity?

I know that, if a matrix commutes with all matrices, then it is a multiple of the identity; see here. The same conclusion holds if a (special) orthogonal matrix commutes with all (special) orthogonal ...
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Does this property of the anti-symmetric matrices hold in general?

Consider an antisymmetric matrix $A=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}$ and an arbitrary vector $v= \begin{pmatrix} a \\ b \end{pmatrix}$. It follows that $(A v)^T v = -ab+ab =0$. My ...
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Find the class of matrices

Find the class of matrices for square matrix $A$ such that Note all computation is over field $Z_3$ $A$ is skew-symmetric and $A^2+ I$ is non-singular, I want to know about the class of such matrices, ...
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Definition of skew-adjoint operator

Let $n\in\mathbb N$. It is customary to call an $n\times n$ matrix $A$ self-adjoint iff the (complex conjugate of the) transpose of $A$ is equal to $A$, and to call $A$ skew-adjoint iff the (complex ...
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The Gram-Schmidt method for a skew-symmetric bilinear form

The book of Hoffman and Kunze, Linear Algebra, section 10.3 Thm. 6 describes a method to obtain a representation of a skew-symmetric bilinear form as a block matrix in the form $$ \begin{bmatrix} J &...
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1answer
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Can the product of a positive definite and a symmetric matrix by skew-symmetric?

I am interested in properties of the matrix product $AB$ where the matrix $A\in\mathbb{R}^{n\times n}$ is positive definite (i.e., $(Av,v)>0$ for all non-zero vectors $v\in\mathbb{R}^n$) and where ...
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determinant in tensor notation

I'm reading Pavel Grinfeld's book "Introduction to tensor analysis and the calculus of moving surfaces". I've reached the chapter where the author talks about determinants; he starts using ...
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Controllability wave equation

Consider the following wave equation, \begin{align} \label{ecuaciononda} z_{tt} - z_{xx} &= 0\qquad < t< T, 0<x<\pi,\\ z(t,0) &= 0\qquad 0 < t < T,\\ (z(0,\cdot),...
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Is the time derative of the rotation vector the angular velocity?

Some Preliminaries: The rotation vector $\phi \in \mathbb{R}^3$ can be converted to rotation matrix $\mathbb{R}^3 \in SO(3)$ by the Rodrigues formula: $$R(\phi) = I + \frac{\sin\theta}\theta \phi_\...
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Help Understanding Proof for Relationship between SO(3) and so(3) Matrices

I am currently reading through a book titled "Modern Robotics," and in it, I have encountered a proof of a proposition stating: Given any $\omega \in \Bbb R^3$ and $R∈SO(3)$, the following ...
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1answer
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Matrix multiplication of a symmetric and skewsymmetric matrix

Consider a matrix multiplication of the form $A^iB^{ij}C^{jk}D^k$, with repeated indices summed. Here $B^{ij}$ and $C^{jk}$ are respectively symmetric and skewsymmetric. Does this product vanish?
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Inverting $T\left(x\right)=Ax+xA=Ax-\left({Ax}\right)^T$ where x and $T(x)$ are antisymmetric and A is symmetric.

I’m currently trying to find $T^{-1}\left(\widetilde{x}\right)$ so I can invert the linear transformation. Equivalently, solving $\widetilde{y}=\widetilde{A}\widetilde{x}+\widetilde{x}\widetilde{A}=\...
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Is there a name for "multiplicatively skew-symmetric matrices" and what are good techniques for computing their determinant?

In the course of a counting problem related to graph paths, I encounter a type of matrix that satisfies the following properties: All diagonal elements are zero, that is, $a_{ii} = 0$ for all i ...
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If $T(v) \otimes w + v \otimes S(w)=0$ are the skew-symmetric operators $T$ and $S$ zero?

Let $V,W$ be finite-dimensional real inner product spaces, and let $T \in \text{Hom}(V,V), S \in \text{Hom}(W,W)$ be skew-symmetric operators. Define the following map $\phi \in \text{Hom}(V\otimes W,...
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Find all $x\in\mathbb{C^n}$ such that $\Im ( x^*Ax )=0$ and $||x||=1$.

Given $A\in\mathbb{C}^{n\times n}$, find all $x\in\mathbb{C^n}$ such that $\Im ( x^*Ax )=0$ and $||x||=1$. My attempt: Any complex number can be decomposed to hermitian part and skew hermitian part: $...
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56 views

Maximum eigenvalue of a skew symmetric matrix

There is a nice method to find the maximum eigenvalue of a real symmetric matrix: Let $A$ be a real symmetric $n\times n$ matrix. Then the maximum eigenvalue of $A$ is given by, $$\lambda_{\max}=\...
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Equivalence of definitons of the pfaffian (via permutations and exterior algebra)

I have worked with the definition of the Pfaffian given in terms of the exterior algebra: Let $E$ be an even dimensional vector space ($dimE=2n$) equipped with a Euclidean metric $g$. Let $f:E \...
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Show that $(A-I)^{-1}(A+I)$ is orthogonal if $A$ is skew-symmetric

In this question, I saw the proof, but I don't get it. $$\begin{aligned} (A - I)^{-1}(A + I) \left( (A - I)^{-1}(A + I) \right)^T &= (A - I)^{-1} (A + I) (A^T + I)(A^T - I)^{-1} \\ &= (A - I)^{...
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$G=\left \{T\in GL(n,\Bbb{R})|T^t ST=S,\text {for all skew symmetric} \space S \in M(n,\Bbb{R})\right \}$ is a subgroup of $GL(n,\Bbb{R})$.

let $G=\left \{T\in GL(n, \Bbb{R}) |T^t ST=S, \text {for all skew symmetric} \space S \in M(n, \Bbb{R})\right \}$ then show that $G$ is a subgroup of $GL(n, \Bbb{R})$. $\textbf{Try}:$ Clearly $I_n \in ...
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About eigenvectors of sorted skew-symmetric Toeplitz matrices

I was playing around with Toeplitz matrices, specifically skew-symmetric Toeplitz matrices. So the diagonal is a zero, every diagonal above (resp. below) the main diagonal is a negative of its ...
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Question on invertible matrices $(E-A)$

Let $A$ be a $n^{th}$ order square and skew-symmetric matrix, if $(E-A)$ is an invertible matrix show that $(E+A)(E-A)^{-1}$ is an invertible matrix (where $E$ is an identity matrix) This is a part of ...
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Relationship between the Pfaffian and determinant from exterior algebra

I have been trying to prove the well-known relation linking the Pfaffian and the determinant from their definitions in terms of the exterior algebra, but unfortunately I haven't been able to work it ...
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How to decompose a $3\times 3$ orthoginal matrix back into Reflection and Rotation components.

I am writing some CAD software that keeps track of the orientation of the modeled components. The oriention is stored as a single $3\times 3$ orientation matrix that is a product of a reflection $R$ ...
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How do I find $\operatorname{det} T_Q$?

Let $S$ be the space of all $n \times n$ real skew symmetric matrices and let $Q$ be a real orthogonal matrix. Consider the map $T_Q: S \to S$ defined by $$T_Q(X) = QXQ^T.$$ Find $\operatorname{det} ...
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What are the practical implications of the eigenvalues of a skew-symmetric matrix being purely imaginary or zero?

I analyse real-world data by decomposing asymmetric square matrices with zero diagonals into a symmetric and a skew-symmetric part, and treating the eigenstructure of the skew-symmetric part as ...
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Given linear operator $A: R^3 \to R^3$, $\forall x \in R^3: (Ax, x)=0$, prove that A is skew-symmetric operator.

Given linear operator $A: R^3 \to R^3$, $\forall x \in R^3: (Ax, x)=0$, prove that A is skew-symmetric operator. I came up with a solution like: if $(Ax, x) = 0 \implies (x, A^{T}x)=(A^{T}x, x)=0 \...
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How can we solve $M^t (dM/dx) = C$?

We have $M[x]$ and $C$ as $3 \times 3$ matrices. If $M^t (dM/dx) = C$, and $C$ is a constant skew-symmetric matrix. How can we solve for $M$? Boundary conditions: $M =$ Identity matrix at $x$ =0. You ...
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Skew-symmetric and symmetric matrices

Let $\mathbb{K}$ be a field with $1 \neq -1$. The set of $n \times n$ skew-symmetric matrices is $\tilde{S}_{n} = \left\{ A \in \operatorname{Mat}_n \;\middle|\; A^T = - A \right\}$ a) prove that $\...
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When do we have orthogonal bases in a vector space equipped with a skew-symmetric form?

Suppose we have a vector space $V$ over some field $\mathbb F$ together with a skew-symmetric form $\langle\space ,\space \rangle$, i.e., $\langle\space ,\space \rangle$ is bilinear and $\langle v,v\...
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What is an example of a skew-symmetric integral transform with given nullity?

What is an example of an integral transform $T_K:L_2(\mathbb{R}^n)\rightarrow L_2(\mathbb{R}^n)$ defined $T_K(f) := \int_{X}K(x,y)f(x)dx$ where $K$ is skew-symmetric ($K(x,y)=-K(y,x)$) which has a ...
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Find number of skew-symmetric matrices of order $3\times 3$ in which all non-diagonal elements are different

Find number of skew-symmetric matrices of order $3\times 3$ in which all non-diagonal elements are different and belong to the set $\{-9,-8,-7,...,7,8,9\}$ My Attempt: I did a simple calculation and ...
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Under what conditions can we find a basis of a Lie algebra such that the adjoint representation acts by skew-symmetric matrices?

My question might be very silly as I've only recently started to learn about Lie algebras. Given a Lie algebra $\mathfrak{g}$, and a vector space basis $\{e_i|i=1 \ldots \dim \mathfrak{g}\}$, we can ...
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Showing results regarding Cayley Transform

Let $\Xi$ be the set of all real $n\times n$ which do not have $1$ as an eigenvalue. Consider the mapping \begin{equation}\phi:\Xi\rightarrow\mathbb{R}^{n\times n},\quad \phi(A)=(A-I)^{-1}(A+I).\end{...
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Prove that any tensor space is the direct sum of the subspaces of the skew-symmetric tensors and of the symmetric tensors.

Definition If $A$ is a pure covariant tensor and $\sigma$ is a given permutation of $r$ elements then we can define a new tensor $T_\sigma A$ by the formula $$ [T_\sigma(A)](\vec v_{_1},...,\vec v_{_r}...
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If matrix $B$ is skew-symmetric with eigenvalue $r$, then $-r$ is also an eigenvalue of $B$

I've considered the determinants of both $B$ and $-B^T$, and found that $\det(B) = (-1)^n \det(B)$, where $B$ is a $n \times n$ matrix. I've also tried the approach where $r$ is an eigenvalue of $B$ ...
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Let $A\in\mathbb{R}^{n\times n}$ be a skew-symmetric matrix, is the Rayleigh quotient of $A$ always $0$?

Let $A\in\mathbb{R}^{n\times n}$ be a skew-symmetric matrix, is the Rayleigh quotient of $A$ always $0$? The Rayleigh quotient is define as following: $$\mathcal{W}_{\mathbb{R}}(A)=\big\{ \frac{x^*Ax}{...
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Relationship determinant symmetric matrix and skew-symmetric counterpart

Say we have a square matrix A and we write it as the sum of its symmetric and skew-symmetric counterpart. Is there any formula which relates the determinants of A, its symmetric and skew-symmetric ...
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$(I-B)^{-1} (I+B)$ always orthogonal if $B$ anti-symmetric

I saw this claim in Strang Intro to linear algebra 5th edition, page 336, and am finding it surprisingly difficult to prove. The claim is that $A = (I-B)^{-1} (I+B)$ is orthogonal if $B^T = -B$. We ...
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Find the Lie algebra from rotation matrix using MATLAB

Let $\omega\in \mathbb{R}^3$, its skew-symmetric matrix is $[\omega]_{\times}$, we know that the corresponding rotation matrix is $\text{exp}([\omega]_{\times})$. My question is: For any rotation ...
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Prove that in a real inner product space $V$ an endomorphism $f$ is skew-Hermitian if and only if $\langle v,f(\vec v)\rangle=0$ for all $v\in V$

Definition Let be $V$ and $U$ real vector spaces equipped with an inner product. So given a linear transformation $f:V\rightarrow U$ a function $f^*:U\rightarrow V$ is called the adjoint of $f$ if $$ \...
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Is there a closed form solution for this problem?

I'm looking for some help on a linear algebra problem that ultimately can be simplified to as follows. Thanks in advance! Let $J$ be an arbitrary full-rank 3 by 3 matrix and $n$ an arbitrary unit 3-d ...
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Unclear statement on symmetric/skew symmetric matrices

There's a statement in the textbook that says; If AB = BA and A and B are symmetric (skew-symmetric), then AB is symmetric. It looks like it means "symmetric or skew-symmetric". Is there a ...