# 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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### 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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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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### 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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### 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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### 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 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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### 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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### 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  \...
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 ...