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DrDu

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I think every normal matrix can be written as A+iB where A and B are commuting hermitian matrices.

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That's a really good idea, thanks a lot!!

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Yes, that's a special case in ##M_1(\mathbb{C})!!##

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And it can be generalized! Diagonal matrices with all real entries are self-adjoint, with complex entries are normal. Every normal operator can be diagonalized with unitary operators as transition matrices, so the general form of a self-adjoint matrix is ##UDU^*## with ##U## unitary and ##D## a diagonal matrix with real entries. The general form of a normal matrix is ##UDU^*## with ##U## unitary and ##D## a diagonal matrix with complex entries.Yes, that's a special case in ##M_1(\mathbb{C})!!##

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