Questions tagged [skew-symmetric-matrices]

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

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Show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric

In a linear algebra textbook, I was given the following problem: If $B$ is a $n \times n$ square matrix, show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric. I know that there are relatively ...
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If a skew-symmetric real matrix has all eigenvalues zero, must it be the zero matrix?

This can be easily verified for $2\times2$ and $3\times3$ matrices, but can the result be generalised?
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Determinant of a $2 \times 2$ block matrix whose diagonal blocks are skew-symmetric

Let $$A = \begin{pmatrix} B & C \\ C' & D\end{pmatrix}$$ be an odd order matrix. If blocks $B, D$ are skew-symmetric matrices, then $\det A=0$. My attempt Without losing generality, we assume ...
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Trace of a power of a skew-symmetric matrix

How to express ${\rm Tr}(A^n)$ (in terms of ${\rm det}\,A$), where $A$ is a skew-symmetric $m\times m$ matrix? With references if possible.
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Justification for $\operatorname{dim} \wedge^k(V)$ is $n\choose k$ the alternating $k$-linear form.

I was told that: $\operatorname{dim} \wedge^k(V)$ is $n\choose k.$ and I am trying to justify this by computing it for different dimensions of $V.$ If $\operatorname{dim}(V) = 1,$ then $v_1 \wedge v_1 ...
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Name for an antisymmetric matrix of signs

Does the antisymmetric matrix $A_{ij} = \text{sign}(i - j)$ have a special name in the literature?
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Prove that the augmenting skew symmetric matrix has full rank.

Suppose that $\vec{v}$ is a unit vector where $v_3 \neq 0$, and $M(\vec{v})$ be skew symmetric matrix written as \begin{equation} M(\vec{v})=\begin{bmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & ...
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How is the spectral decomposition of a skew-symmetric matrix $A$ related to that of $iA$?

Any skew-symmetric matrix can be written as $A=UQU^\dagger$ where $U$ is unitary and \begin{align*} Q=\begin{bmatrix} 0 & \lambda_1 & \\ -\lambda_1 & 0 & \\ & & ...
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Skew-Symmetric and Upper-Triangular Real Matrices forming a Direct Sum

I am working on trying to prove that $$\text{Mat}(n,\mathbb{R}) = \mathcal{S} \oplus \mathcal{T}$$ where $\mathcal{S}$ is the set of all skew-symmetric, real matrices of size n and $\mathcal{T}$ is ...
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Finding matrix $\mathbf{Q}$ that satisfies $\mathbf{J=Q^TAQ}$ of skew-symmetric $\mathbf{A}$ and $\mathbf{J}$

Let $\mathbf{A}$ and $\mathbf{J}$ be real, full-rank, skew-symmetric, square matrices. If we know those matrices, can we find a real square matrix $\mathbf{Q}$ that satisfies $\mathbf{J=Q^TAQ}$? ...
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Calculating Skew Symmetric Matrix determinant?

I've been solving matrix related questions and I'm confused for this one: If A is a 5 * 5 matrix, where A^t= - A (A transpose equal to negative A) then what is the determinant for A? There are 4 ...
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Doubts on symmetric and a skew-symmetric matrix

Given two following statements: $1.$ "The diagonal elements of a skew-symmetric matrix are all zero." $2.$ "A real/complex square matrix can be uniquely expressed as the sum of a ...
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SU(2) invariant structures and the Frobenius Schur indicator

For $SU(2)$, the fundamental representation is a quaternionic representation. Which means there is a preserved skew symmetric form, written as a matrix: $$ \varepsilon = \left(\begin{array}{cc} 0 &...
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Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have a very specific question concerning the Fourier transform of certain determinant function of a skew-symmetric matrix: Write $SSym(n)$ for the space of $n\times n$ skew-symmetric real matrices. ...
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Modified skew-symmetric matrix eigenvalues

I know that the eigenvalues of a real skew-symmetric matrix $A$ are either zero or imaginary. I have the following modification: let $C$ be a diagonal matrix with ones for some diagonal elements and ...
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Is the determinant of a skew symmetric matrix nonzero if nondiagonal entries are all nonzero?

Let $A = [a_{i,j}]_{i,j} \in M_{2n}(\mathbb{R})$ be a skew-symmetric matrix such that $a_{i,j} \neq 0$, $\forall i \neq j$. I am trying to see if it's true that $\det A \neq 0$. I tried working from ...
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Solution to a linear equation involving a skew-symmetric tensor

Say that $S(\mathbf{x})$ is a skew-symmetric $k+1$-tensor, that is, $S_{i_0,...,i_a,...,i_b,...,i_{k}}(\mathbf{x})=-S_{i_0,...,i_b,...,i_a,...,i_{k}}(\mathbf{x})$ for $a,b=0,...,k$, then find $S(\...
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Prove that the set of skew-symmetric matrices is closed under addition

I am trying to prove that W is a subspace of V with: $V = M_{n\times n}$, $W = \{A \in M_{n\times n} : A = -A^T\}$ I am fairly sure $W$ is closed under addition, but am not sure how to prove it for ...
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Skew symmetric tensor

I am currently working through Tensor calculus and differential geometry by Prasun Nayak, however I am confused where something with skew symmetric tensors has came from. In the last line I am ...
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Is there are relationship between the eigenvectors and the real Schur vectors of a real skew-symmetric matrix?

A real skew-symmetric matrix $A$ can be diagonalized with complex eigenvectors and pure imaginary eigenvalues: $$A=V S V^*$$ where $S$ is: $$S = \begin{pmatrix} -\lambda_1\mathrm{i} & 0 & 0 &...
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Suppose $A$ is a $ 3 × 3$ matrix and it satisfies $ A^T = −A$. Prove that $\det(A)= 0$. [duplicate]

I think it may relate to the space? $A^T = -A$ can prove the column space is equal to the row space. How to prove the $\det(A)$?
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How to find a symplectic basis

We know that corresponding to any skew-symmetric bilinear form $f$, there exists a basis with respect to which the matrix of $f$ will look like $$\begin{pmatrix}J& &\\ &J& \\ & &...
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Every real, skew-symmetric matrix is diagonalisable by a unitary matrix

I need to show, that every real, skew-symmetric matrix M can be diagonalized by a unitary matrix U. $$ M=-M^T \implies M = U D U^\dagger \quad \textrm{with} \quad U U^\dagger = U^\dagger U = \mathbb{I}...
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Skew Symmetric Matrix with condition

If A is 2*2 skew symmetric matrix with $$A^2=A$$, Then A=0 $$$$ My attempt was by assuming the matrix A and substituting the squaring condition and the condition of skew symmetric and got system of ...
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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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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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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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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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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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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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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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