# Questions tagged [skew-symmetric-matrices]

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

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### Anti-intuition kurtosis

I want to describe the tailedness of distribution and calculated the kurtosis of "before" and "after", and it seems like "after" has a larger kurtosis than "before&...
1 vote
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### A general solution $M$ for the matrix equation $A^* M^* + M A = 0$

As titled, for some unknown complex-valued matrix $A$, possibly rectangular, does there in general exist a complex-valued matrix $M$ such that $$A^* M^* + M A = 0$$ Here, $A^*$ denotes the ...
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1 vote
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### Skew-symmetric matrix least-square problem.

Is there a simple solution to the following problem: Given $x,y\in\mathbb{R}^n$, $$\min_{A\in M^n(\mathbb{R})} \left\| \left(\frac{A-A^t}{2}\right)x - y\right\|^2$$ Or equivalently $\min ||Ax-y||^2$...
138 views

### $\det(A^2-B^2) \leq \det(A^2)$ when $B$ is of rank 1

This is an exercise problem given at linear algebra class. As a problem before this I was able to show that $$\det(A + uv^T) = \det A + v^T (adj A )u$$ when $A$ is order $n$ square matrix and $u,v$ ...
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### Show that $A\in M_n(\mathbb R)$ is skew-symmetric if $\det(A+M)=0$ for all M skew-symmetric, with $n$ odd.

Let $n$ be an odd integer, and $A\in M_n(\mathbb R)$ such that, for all $M\in M_n(\mathbb R)$ skew-symmetric, we have: $$\det(A+M)=0$$ Show that $A$ is skew-symmetric. What I did so far: Nothing much ...
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### Can the product of a complex symmetric unitary matrix and a skew-hermitian matrix be complex skew-symmetric?

Let $S$ be a nonzero complex symmetric unitary matrix ($S\neq 0$, $S = S^T$, $S^HS=Id$) Let $B$ be a nonzero skew-hermitian matrix ($B\neq 0$, $B^H = -B$) Can their product $P\triangleq SB$ be ...
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1 vote
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### For skew-symmetric $U$, prove that $U(1-U^\dagger U)^{-1}$ is, too

Let $U$ be a complex skew-symmetric matrix ($U = -U^\mathsf{T}$, where $U^\mathsf{T}$ means the transpose of $U$). I want to show that $U(1 - U^\dagger U)^{-1}$ is also skew-symmetric. Here $U^\dagger$...
37 views

### Definition of the (anti-)symmetric part of a tensor

Consider the following tensor: $$A_{ij} = B_{k,i}C_{k,j}$$ which can be decomposed into its symmetric and antisymmetric parts: $A_{ij} = A_{ij}^s + A_{ij}^a$. I'm all confused as to how to express ...
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61 views

1 vote
### 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 ...