Questions tagged [skew-symmetric-matrices]

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

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Proving symplectic identity

Let $\Lambda$ be a skew-symmetric matrix and $Q$ a symmetric matrix. Let $\text{Id}$ be the identity matrix and $h > 0$ a real number. I am trying to prove the following identity: $$ (\text{Id} + \...
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Eigenvalues of symmetric and skew-symmetric zero line sum matrix

Suppose I have a $n \times n$ symmetric matrix along the main diagonal that is also anti-symmetric along the other diagonal. The rows and columns add to 0. For example, $$ \begin{bmatrix} a & b &...
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A real skew symmetric matrix of order $3$ is diagonalizable over $\Bbb{C}$

Let $M$ be a $3$ $\times$ $3$ skew symmetric matrix with real entries. Then I need to show that $M$ is diagonalizable over $\Bbb{C}$. This has been my attempt. The characteristic polynomial will be of ...
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1answer
34 views

Show the proof of this property of skew-symmetric matrix?

For a vector $T \in \mathbb{R}^3$ and a matrix $K \in \mathbb{R}^{3x3}$, if $det(K) = +1$ and $T' = KT$, then $\hat{T} = K^T \hat{T'} K $, where the hat operator denotes the skew-symmetric matrix ...
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1answer
13 views

skew-diagonalizing an anti-symmetrc matrix

Let's assume that i have a (real) $2N\times2N$ anti-symmetric matrix $B=\left\{ b_{ij}\right\} $ with the property that $BB^{T}=\boldsymbol{1}$ where $\boldsymbol{1}$ is the identity matrix. Is it ...
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1answer
33 views

Establish collinearity of two vectors

I'm struggling in proving collinearity of two vectors provided that the relations described here below hold. Also, I was wondering if such condition can be relaxed. Consider a vector $x\in\mathbb{R}^n$...
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29 views

Positive vector in the kernel of an antisymmetric “adjacency” matrix

For $n\in\mathbb{N}^*$, let $V=[[1,n]]$, and $G=(V,A)$ be an oriented graph, stronlgy connected with $n$ vertices. Let $M\in\mathcal{M}_n(\mathbb{R})=(m_{i,j})$ be an skew-symmetric matrix of size $n$ ...
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Show the correctness of this identity?

I'm studying the Lie theory, and this is an identity presented in a paper dedicated to introducing the essential of the Lie theory. Interpretation of the symbols: R, the rotation matrix. $\theta$, ...
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25 views

Levi-Civita with binary multi-index

Consider the Levi-Civita symbol $\varepsilon_{ijkl}$ where $i,j,k,l \in \{0,1,2,3\}$. Consider now the binary representation of $i,j,k$ and $l$, such that $i \to (i_0 \, i_1)$, $j \to (j_0 \, j_1)$, $...
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1answer
35 views

Why does adding up all the entries of an adjoint matrix of a 2n x 2n sized skew-symmetric matrix equal 0

I'm in the middle of proving this statement. If $A$ is a $2n \times 2n$ sized skew-symmetric matrix, then $det(A) = det(A + xJ)$ whereas $J$ is an $2n \times 2n$ sized matrix with all of its ...
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Relation between eigenvectors and singular vectors of complex skew-symmetric matrices

As shown in this answer, if $A$ is a real skew-symmetric matrix, and $v,w$ are a pair of orthogonal singular vectors with $$Av=sw \qquad\text{ and }\qquad Aw=-sv,$$ for some $s>0$, then the ...
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Prove that Let A be a definite positive matrix, such that for every positive integer k, there exists a symmetric matrix B such that A=B^k

I need some help, I need to prove the following exercise: Let A be a definite positive matrix, such that for every positive integer k, there exists a symmetric matrix B such that A=B^k I haven't ...
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1answer
56 views

Solving matrix quadratic equation [closed]

Let $\mathbf{A}, \mathbf{C}\in \mathbb{R}^{n\times n}$. Given a skew-symmetric matrix $\mathbf{G}$ I am looking for any numerical procedure so solve the following quadratic equation. $$ \mathbf{B}^\...
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How to find the parametric equations for the line through the point that is parallel to the plane.

How to find the equations of the line passing through the point $(8, 7, 9)$, intersecting the $z$-axis and parallel to the plane $2x−6y+z−12=0$.
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Show that the cross product a x b is skew symmetric.

Given two vectors a, b ∈ R3: a x b (cross product). Show that the cross product is skew symmetric. If AT = −A which means A is skew symmetric then prove that (A+B) is also skew symmetric. I managed to ...
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1answer
39 views

Is there a relationship between the standard vector cross product and the vector cap product?

I just finished reading through Introduction to Matrices and Vectors, International Student Edition, by Jacob T. Schwartz. In chapter 6 they proposed a definition I've never seen before, that seems ...
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Diagonal elements and determinant of an antisymmetric matrix

The question is the next: Show that the elements of the diagonal of an antisymmetric matrix are 0 and that its determinant is also 0 when the matrix is ​​of odd order. I have shown in a previous ...
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1answer
33 views

product of skew-symmetric and symmetic matrix: diagonal elements

This is a second attempt, related to my earlier question zero diagonal of product of skew-symmetric and symmetric matrix with strictly positive identical diagonal elements where I think I asked the ...
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1answer
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zero diagonal of product of skew-symmetric and symmetric matrix with strictly positive identical diagonal elements

I encountered the following problem in a proof which I couldn't solve so far: Let $\mathbf{A}$ be a skew-symmetric matrix, and $\mathbf{B}$ a symmetric matrix. There is at least one off-diagonal ...
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Is there an analog of polarization for skew-symmetric forms?

Polarization works both ways. Not only can you represent any homogeneous polynomial $f$ of degree $d$ as $F(x,...,x)$ for a multilinear form $F(x_1,...,x_d)$ but also conversely, any symmetric ...
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Find an invertible matrix for this equality

Let $K$ be a field with $char(K) \neq 2$. Let $S$ be an invertible $n \times n$ matrix where $n>1$. Show that there exists an invertible matrix $A$ such that $A^TS+SA=0$. I am stuck with this ...
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73 views

Product of a symmetric and anti-symmetric matrix

I have the following question about matrices, Let $S$ and $A$ be two $n \times n$ matrices which are respectively symmetric and anti-symmetric. Can I conclude anything about the products $SA$ or $AS$, ...
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Relationship between columns of a unitary matrix

$U=[u_1\quad u_2\quad u_3]$ is a 3 x 3 unitary matrix and $u_i\in \mathbb R^3, i=1,2,3$. It seems that $$u_2u_1^T - u_1u_2^T$$ is equal to $[u_3]_\times$, i.e. the skew-symmetric matrix of $u_3$ (up ...
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Common neighbour matrix of a graph

Problem Let $G = (V,E)$ be a simple graph, $A$ its adjacency matrix and let $c(u,v) = |N(u) \cap N(v)|$ be the number of common neighbours of any pair of nodes $u,v \in V$, i.e. $$c(u,v) = \big|\big\{...
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Find an ordered basis $\mathcal{B}$ for $M_{2\times 2}(\textbf{R})$ such that $[T]_{\mathcal{B}}$ is a diagonal matrix.

Let $T$ be a linear operator on $M_{n\times n}(\textbf{R})$ defined by $T(A) = A^{t}$. (a) Show that $\pm 1$ are the only eigenvalues of $T$. (b) Describe the eigenvectors corresponding to each ...
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Orthogonality of skew-symmetric matrix

Let $S$ be a real skew-symmetric matrix. Prove that $I-S$ and $I+S$ are orthogonal, where $I$ is $n \times n$ identity matrix. My approach: I am not sure whether the question is asking to prove I-S ...
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1answer
59 views

Prove that a skew symmetric matrix has at least one eigenvalue that $||\lambda_{\text{max}}||_2 > 1$

Assume that we have a skew symmetric matrix $A^T = -A$ and we want to prove that this matrix $A$ has at least one eigenvalue $||\lambda_{\text{max}}||_2 > 1$. I have tried power iteration ...
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5answers
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Proving Invertibility of Matrices

Given that $A$ is an $n×n$ skew symmetric matrix $I$ being the $n×n$ identity matrix, prove that $A − I$ and $A + I$ are invertible. Can trace of $A$ be non-zero number ? Any suggestions how to go ...
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Centralizer of $\mathfrak{so}_{n}(\mathbb{R})$

I want to compute the centralizer of $\mathfrak{so}_{n}(\mathbb{R})$ as a subalgebra of $\mathfrak{gl}_n(\mathbb{R})$: $$ C_{\mathfrak{gl}_n}(\mathfrak{so}_{n}(\mathbb{R}))=\{X\in \mathfrak{gl}_{n}(\...
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Conditions for $e^X$ to be an orthogonal matrix [duplicate]

If $X$ is a skew-symmetric matrix, its exponential $e^X$ is orthogonal. Does the converse implication hold?
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Proof the following properties of antisymmetric matrices:

a) If A is antisymmetric of order $p$, then $\det(A)=(-1)^pdet(A)$ b) If A is odd order antisymmetric, then $\det(A)=0$ Remembering that if A is antisymmetric, $A ^ T = -A$ How could I prove the ...
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1answer
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The boundary of the convex hull of squares of skew-symmetric matrices

Let $n \ge 3$, and let $C$ be the convex cone generated by the squares of all real $n \times n$ skew-symmetric matrices. Is $C$ closed in $\mathbb{R}^{n^2}$? What is its boundary? $C$ is a strictly ...
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Does every negative semidefinite matrix lie in the convex cone generated by the squares of skew-symmetric matrices?

Let $C$ be the convex cone generated by all the squares of real $n \times n$ skew-symmetric matrices. Does every negative-semidefinite matrix lie in $C$? I know that every square of a skew-...
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Skew-symmetric matrix decomposition with polynomials

I would like to ask for some help on the following problem: Suppose that $A$ is an antisymmetric (or skew-symmetric) complex $2n\times 2n$ unitary matrix, i.e. such that $AA^* = A^* A = I_{2n}$ and $...
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The squares of skew-symmetric matrices span all symmetric matrices

This is a self-answered question. I post this here, since it wasn't obvious for me at first, and I think it might be helpful for someone at some future time (maybe even future me...). Claim: Let $n ...
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Find the rank of the matrix $M$

Let $M$ be the $(2n+1)\times (2n+1)$ skew symmetric matrix with entries given by $$a_{ij} = \begin{cases} \,\,\,\, 1 & \text{if } i-j\in \{-2n, -2n+1, \dots, -n-1\} \cup \{1, 2, \dots, n\}\\ -1 &...
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Sum of squares of skew-symmetric matrices which is not a square itself

This is just a curiosity. How can I find an example of two real $n \times n$ skew-symmetric matrices $A,B$, whose sum of squares $A^2+B^2$ is not a square of any skew-symmetric matrix? The skew-...
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4answers
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If A is a matrix such that $A^{2}+A+2I=O$ ,then $A$ can't be skew symmetric.

If $A$ is a matrix such that $A^{2}+A+2I=O$, then $A$ can't be skew symmetric. (True/false) When $A$ is odd order matrix then the statement is true, since $A$ is non singular. $( |A||A+I| = (-2)^{n})$...
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1answer
910 views

Adjoint of skew-symmetric matrix

If $A$ is a skew-symmetric $n\times n$ matrix, verify that $\operatorname{adj} A$ is symmetric or skew-symmetric according to whether $n$ is odd or even. Things I can think of is $A^T=-A$ for skew-...
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Skew-symmetric square root of symmetric matrix

Suppose that $A$ and $B$ are real skew-symmetric $4 \times 4$ matrices. Hence $A^2$ and $B^2$ are symmetric matrices. Now we want to find a skew-symmetric 4×4 matrix ($C$) which satisfies $$A^2+B^2=C^...
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Skew-Symmetric vs Symmetric

If $A$ is a symmetric $n × n$ matrix and $B$ is a skew symmetric $n × n$ matrix, which of the following are true? (a) $ABA$ is symmetric (b) $ABA$ is skew-symmetric (c) $AB^2A$ is symmetric (d) $AB^2A$...
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Why are skew-symmetric matrices of interest?

I am currently following a course on nonlinear algebra (topics include varieties, elimination, linear spaces, grassmannians etc.). Especially in the exercises we work a lot with skew-symmetric ...
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Congruence of skew-symmetric matrices

Prove that every matrix congruent to a skew symmetric matrix is skew symmetric. My work- How do I prove that the transforming matrix $P$ is always orthogonal? Please help.
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Is it possible to have a $3 \times 3$ matrix that is both orthogonal and skew-symmetric?

Is it possible to have a $3 \times 3$ matrix that is both orthogonal and skew-symmetric? I know it has something to do with the odd order of the matrix and it is not possible to have such a matrix. ...
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3answers
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Is every odd order skew-symmetric matrix singular?

We call a square matrix $A$ a skew-symmetric matrix if $A=-A^T$. A matrix is said to be singular if its determinant is zero. Is every odd order skew-symmetric matrix with complex entries singular?
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Equivalence of skew-symmetric matrices

Let $N=\{1,\dots,n\}$ and $A,B$ be $n\times n$ skew symmetric matrices such that it is possible to permute some rows and some columns from $A$ to get $B$. In other words, for some permutations $g,h: N\...
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295 views

Skew-symmetric matrix property

This page gives the relation $\left[R\vec{\omega}\right]_{\times}=R\left[\vec{\omega}\right]_{\times}R^{T}$ where $R$ is a DCM (Direction Cosine Matrix), $\vec{v}$ is the angular velocity vector and $[...
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2answers
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Invertible skew-symmetric matrix

I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the ...
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1answer
410 views

Skew-symmetry of matrix $C = AB - BA$

Matrices $A$ and $B$ are skew-symmetric and $C = AB - BA$. Show $C$ is also skew-symmetric. I see $\mbox{tr}(C)=0$ and $C = AB - (AB)^T$, but nothing else.
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What is the derivative of a skew symmetric matrix?

I'm trying to work out some Jacobians and I ran across a problem. If I have a function of a vector making it a skew symmetric matrix, like below, what is the derivative $f'$? $$ f(\boldsymbol{\omega})...