# Questions tagged [skew-symmetric-matrices]

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

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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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### 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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### 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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### 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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### 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 $n$ ...
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. $\theta$, ...