Home > Higher Algebra, Olympiad > Characterization of a normal matrix

Characterization of a normal matrix

A n\times n,\ (n \geq 2) matrix A with complex entries is called normal if AA^*=A^*A, where A^* is the conjugate transpose of A. Prove that A is normal if and only if \text{rank}\; (AA^*-A^*A) \leq 1.

Hint: Prove that if C is a matrix of rank 1 then C^2=\text{Tr}(C)\cdot C.

Categories: Higher Algebra, Olympiad Tags: ,
  1. removablesingularity
    April 30, 2012 at 9:31 am

    What happens when rank (AA^* - A^*A) = 1?

    • April 30, 2012 at 10:44 am

      When rank of AA^*-A^*A is 1 use the hint, since the trace of this matrix is zero and therefore (AA^*-A^*A)^2=0. Then if you denote X=AA^*-A^*A and note that X^*=X you get XX^*=X^2=0, and therefore X=0.

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