The set of all $n$-square matrices with trace $0$ is a subspace of the set of all $n$-square matrices. Is there a standard notation and/or name for this subspace?
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Yes. These matrices are called "traceless" or "tracefree", and the subspace comprised of them is called $\mathfrak{sl}_n$, the special linear Lie algebra. This term is used because the traceless matrices form the Lie algebra associated with the special linear Lie group $SL_n$ consisting of all $n$-by-$n$ matrices with determinant $1$. |
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