# 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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### 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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### $G=\left \{T\in GL(n,\Bbb{R})|T^t ST=S,\text {for all skew symmetric} \space S \in M(n,\Bbb{R})\right \}$ 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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