Let A ∈ M_n (C).

Let <*,*> be the standard inner product in C^n, viewed either as row vectors or as column vectors.

Let r_j be the j-th row of A, and let c_j be the j-th column of A.

Show that A is normal, if and only if

<r_i, r_j> = <c_j, c_i>

for all i,j, 1≤i, j≤n.

Is it somehow related to the Spectral theorem? I don't even know how to start this question. Please help! Thanks in advance!!!