Questions tagged [skew-symmetric-matrices]
Matrix $A$ is skew-symmetric (or antisymmetric) iff $A^\top = -A$.
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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 ...
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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 ...
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Find an expression for the nth power of this matrix
I'm looking to derive an expression for $A^n$
Let $A$ = $\begin{pmatrix} 0 & \alpha \\ -\alpha & 0 \end{pmatrix}$
Let $S$ = $\begin{pmatrix} 0 & 1\\ -1& 0 \end{pmatrix}$
Am I correct ...
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3
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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 ...
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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 ...
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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 ...
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How to prove the following vector quadruple cross product using skew-symmetric representation
in the following expression I'm curious to know is there a way to prove the following quadruple cross product:
$$
\vec{a} \times (\vec{b} \times(\vec{a} \times \vec{b}))=\vec{b} \times (\vec{a} \times(...
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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 &...
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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 ...
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Diagonalization of Complex skew symmetric matrices [duplicate]
I am trying to understand if all complex skew symmetric matrices (i.e A=-Transpose(A)) where the entries are all Complex numbers, can be diagonalized.
I have been looking on the internet for a few ...
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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 ...
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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?
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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 ...
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Energy with skew-symmetric matrix, gradient
I have formulated the energy $E(x) = x^\top Bx$ where $B\in\mathbb{R}^{n\times n}$ is skew-symmetric ($B^\top=-B$). I want to find the stationary points of said energy, thus I need to take the ...
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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)$$ Which of the following statements are true?
$1.$ $\|x(t)\|...
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A skew symmetric and orthogonal matrix has eigen values (3/5) + (4i/5). How can this be possible? It must have 0 or purely imaginary values. Problem 1
Problem 1. It is Orthogonal and skew symmetric but eigen values aren't purely imaginary or zero
Are the following matrices symmetric, skew-symmetric and/or orthogonal?
$$\frac15\begin{bmatrix}3&-4\...
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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 ...
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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
$$
M(k)_{ij} =\frac{i-j}{i+j},\,1\le i\le2k,\, \,1\le j\...
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Pfaffian definition for a complex skew-Hermitian matrix?
Normally, the Pfaffian is defined for real skew-symmetric matrices $A = -A^T$, and some authors extend this to complex skew-symmetric matrices. Is there a straightforward generalization of the ...
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Prove that the third partial derivative over the space is a formally skew-adjoint operator in $\mathbb{R}$
I'm trying to solve the following question:
Prove that $\mathcal{J} = \frac{\partial^3}{\partial z^3}$ is formally skew-adjoint on the space $C^ \infty(I, \mathbb{R}) $ of smooth real-valued ...
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Commutator of Skew-Symmetric Matrices with Integer entries
Let $\mathcal{S_3}$ be the collection of $3\times 3$ skew-symmetric matrices with integer entries. There is no non-zero $D\in \mathcal{S_3}$ that satisfies the following equation
$$[D,[D,[D,N]]]=0$$
...
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Is $I-JR$ invertible if $J$ is skew-symmetric and $R$ is symmetric positive semi-definite?
Given any skew-symmetric matrix $J \in \mathbb{R}^{n \times n}$, we know that $(I-J)$ is invertible, where $I \in \mathbb{R}^{n \times n}$ denotes the identity matrix. Now, assume that $J \in \mathbb{...
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Linear transformation of skew-symmetric block matrix
Suppose that we have the following special skew-symmetric block matrix $$ J = \begin{bmatrix} J_1 & -A & -B\\ A^T & J_2 & -C \\ B^T & C^T & J_3 \end{bmatrix} \in R^{(2n+m) \...
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Matrix representation of the skew-symmetric operator
Assume that $A$ is a skew-symmetric (skew-hermitian) operator on the finite dimensional unitary vector space $V$. I'm interested in the matrix representation of this operator. I found that there ...
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inverse Cayley transformation of product of two Cayley transformations
Let two skew-symmetric matrices $A$, $B$ be given.
We can apply the Cayley transformation $A\mapsto (I-A)(I+A)^{-1}$ to get orthogonal matrices which do not have an eigenvalue $=-1$.
Let $\tilde{A}:=(...
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Can we generate a block skew-symmetric matrix by a operator
Suppose we have following speical block matrix
$$
X = \begin{bmatrix}
0 & X_3 & -X_2\\
-X_3& 0& X_1\\
X_2 &-X_1 &0
\end{bmatrix}
$$
where $X1,X2,X3 \in R^{n \times n}$ and not ...
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The rank of block skew symmetric matrix
Suppose we have following speical block skew symmetric matrix
$$
X = \begin{bmatrix}
0 & X_3 & -X_2\\
-X_3& 0& X_1\\
X_2 &-X_1 &0
\end{bmatrix}
$$
where $X_1,X_2,X_3\in R^{n \...
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Antisymmetric Matrices and Orthogonality
My notes state: Given an orthogonal $n\times n$ matrix $A=I+pB$, where $I$ is the identity matrix, $p\ne 0$ is a real number and $B$ is an $n\times n$ matrix, then $B$ is skew-symmetric.
I know that $$...
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Proof that $(A^t)^t=A$, $(A+B)^t=A^t+B^t$, $(AB)^t=B^tA^t$, and deduce that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric
In a linear algebra textbook, I was given the following problem:
If $B$ is a $n \times n$ square matrix, show that $BB^t$ is symmetric and $B-B^t$ is skew-symmetric.
I know that there are relatively ...
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If a skew-symmetric real matrix has all eigenvalues zero, must it be the zero matrix?
This can be easily verified for $2\times2$ and $3\times3$ matrices, but can the result be generalised?
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Determinant of a $2 \times 2$ block matrix whose diagonal blocks are skew-symmetric
Let $$A = \begin{pmatrix} B & C \\ C' & D\end{pmatrix}$$ be an odd order matrix. If blocks $B, D$ are skew-symmetric matrices, then $\det A=0$.
My attempt
Without losing generality, we assume ...
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Trace of a power of a skew-symmetric matrix
How to express ${\rm Tr}(A^n)$ (in terms of ${\rm det}\,A$), where $A$ is a skew-symmetric $m\times m$ matrix? With references if possible.
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Justification for $\operatorname{dim} \wedge^k(V)$ is $n\choose k$ the alternating $k$-linear form.
I was told that: $\operatorname{dim} \wedge^k(V)$ is $n\choose k.$ and I am trying to justify this by computing it for different dimensions of $V.$
If $\operatorname{dim}(V) = 1,$ then $v_1 \wedge v_1 ...
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Prove that the augmenting skew symmetric matrix has full rank.
Suppose that $\vec{v}$ is a unit vector where $v_3 \neq 0$, and $M(\vec{v})$ be skew symmetric matrix written as
\begin{equation}
M(\vec{v})=\begin{bmatrix} 0 & -v_3 & v_2 \\ v_3 & 0 & ...
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How is the spectral decomposition of a skew-symmetric matrix $A$ related to that of $iA$?
Any skew-symmetric matrix can be written as $A=UQU^\dagger$ where $U$ is unitary and
\begin{align*}
Q=\begin{bmatrix}
0 & \lambda_1 & \\
-\lambda_1 & 0 & \\
& & ...
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Skew-Symmetric and Upper-Triangular Real Matrices forming a Direct Sum
I am working on trying to prove that $$\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}...
