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in Matrices & determinants by (41.3k points)

Show that every square matrix A can be uniquely expressed as P + iQ, where P and Q are Hermitian matrices.

1 Answer

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Therefore, P is a Hermitian matrix.

Also

Therefore, Q is also a Hermitian matrix.

Thus, A can be expressed in the form (1).

For A to be unique, let A = R + iS, where R and S are both Hermitian matrices. We have

Aθ = (R + iS)θ = Rθ + (iS)θ = Rθ + iSθ = Rθ - iSθ

= R - iS (since R and S are both Hermitian matrices)

Therefore,

A + Aθ = (R + iS) + (R - iS) = 2R

⇒ R = 1/2(A + Aθ) = P

Also,

A - Aθ = (R + iS) - (R - iS) = 2iS

⇒ S = 1/2i(A - Aθ) = Q

Hence, expression (1) for A is unique

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