Compare eigenvalues of the Normal matrix A and the matrix (A+A^T)/2

Let $A$ a $nxn$, non-symmetric, real, Normal matrix with pairs of complex conjugate eigenvalues (and at least one real eigenvalue). I find, through Maple for any $n>2$, that the matrix: $B:=(A+A^{T})/2$ has eigenvalues the Real part of the complex eigenvalues of the matrix $A$ and the same real.

I wonder if anyone has encounter this property in bibliography before. If so i would be grateful for a reference to find the corresponding proof.

flag	asked Feb 17 at 12:10 GiorgosKan 1

This is not a research level question, but: If $A$ is real and normal and $Av = \lambda v$ for some non-zero vector $v \in \mathbb{C}^{n},$ then we also have $A^{t}v \overline{\lambda}v,$ and $v$ is an eigenvector of $\frac{A+A^{t}{2}$ with eigenvalue ${\rm Re}(\lambda).$ For

$0 = \langle (A- \lambda I)v,(A - \lambda I)v \rangle = \langle (A^{t} - \overline{\lambda}v)(A^{t} - \overline{\lambda})v \rangle $

, using normality. – Geoff Robinson Feb 17 at 14:37

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