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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 symmetric matrix and a skew-symmetric matrix."

As per my knowledge, these two statements are true. But I have a small doubt, Do these two statements hold every time ? or there are some cases where they don't.

Actually I saw somewhere that "these are not true if the ground field F be of characteristic $2$". But they doesn't provide any further details. So if any one has any point about my doubt, please provide me the same.

user26857
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nmasanta
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  • Indeed, by this answer for a field with characteristic 2, statement 1 is not true. – Kurt G. Mar 15 '22 at 10:25
  • I also mention the same in my last paragraph @KurtG. – nmasanta Mar 15 '22 at 10:36
  • So we agree. To show item 1 take a diagonal element $x$ of a skew-symmetric matrix. It must satisfy $-x=x$ which implies $x=0$ unless the field has characteristic 2. – Kurt G. Mar 15 '22 at 10:38
  • So by this property, in the ground field F be of characteristic $2$, $\begin{pmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1 \end{pmatrix}$ is a skew-symmetric matrix. Right ? – nmasanta Mar 15 '22 at 10:48
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    '@nmasanta' : Absolutely. And to make things worse. It is symmetric and skew-symmetric. – Kurt G. Mar 15 '22 at 10:53
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    Is $\pmatrix{1&1\cr0&0\cr}$ the sum of a symmetric and a skew-symmetric over a field of characteristic two? – Gerry Myerson Mar 15 '22 at 11:31
  • In fact with characteristic 2, neither 1 nor 2 is true. – Oscar Lanzi Mar 15 '22 at 12:02
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    @GerryMyerson . No and thanks ! Write $(m_{ij})$ for this matrix. If there existed a symmetric matrix $(s_{ij})$ and a skew-symmetrix matrix $(a_{ij})$ s.t. $m_{ij}=s_{ij}+a_{ij}$ then $1=m_{12}+m_{21}=s_{12}+s_{21}=s_{12}+s_{12}=0$. – Kurt G. Mar 15 '22 at 12:08
  • @OscarLanzi . Agreed . – Kurt G. Mar 15 '22 at 12:09

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