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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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Thanks; I don't know much about Lie algebras, but do you just mean that by consequences of the definitions of Lie group and Lie algebra that it turns out that $\mathfrak{sl}_n$ is a Lie algebra of the lie group $\mathsf{SL}_n$? That's interesting... –  AFX May 13 '12 at 22:45
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