Why is the square of a normal matrix as well a normal matrix?

Question

Or is it? Be $A$ a normal matrix and my question is if $A^2$ is as well a normal matrix?

Have you tried applying the definition? – Qiaochu Yuan May 22 '13 at 21:52 — Qiaochu Yuan, May 22 '13 at 21:52

score 2 · Answer 1 · answered May 22 '13 at 21:53

2

A matrix $A$ is normal iff $AA^*=A^*A$. If $A$ is normal, then $$A^2(A^2)^*=A^2(A^*)^2=AAA^*A^*=AA^*AA^*=A^*AAA^*=A^*AA^*A=A^*A^*AA=(A^2)^*A^2$$ thus $A^2$ is normal as well.

answered May 22 '13 at 21:53

Alex Becker

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Why is the square of a normal matrix as well a normal matrix?

1 Answers1