Relationship between the eigenvalues of a complex matrix and the eigenvalues of its skew-hermitian part

Suppose, $A \in \mathbb{C}^{n \times n}$ is a non-hermitian matrix and it is decomposed as $A = \dfrac{1}{2}[A + A^{\star}] + \dfrac{1}{2}[A - A^{\star}]$. Now if $\lambda_{i}[A-A^{\star}] = -j b \; \forall i$, where $b \in \mathbb{R}^{+}$, then can it be guaranteed that the imaginary part of $\lambda_{i}[A] < 0$ for $i=1,2 \cdots n$ ? For example $A = \left[\begin{array}{cc} -0.4878 - j0.3902 & -0.2439 - j0.1951 \\ -0.2439 - j0.1951 & -0.4878 - j0.3902 \end{array} \right]$ for which $\lambda_{i}[A-A^{\star}] = \{-j0.3902, \; -j 1.1707\}$.

• Is spectral theorem useful in this regard? Please suggest any relevant references. – Parijat Bhowmick Aug 21 '16 at 14:18
• Can you define $j[A - A^*]$? What is this? – IAmNoOne Aug 21 '16 at 14:20
• This means that the eigenvalues of the skew-hermitian part of A matrix are all negative. – Parijat Bhowmick Aug 21 '16 at 14:22
• What is $A$ doing there then? – IAmNoOne Aug 21 '16 at 14:26
• A is any complex matrix; neither hermitian, nor symmetric. – Parijat Bhowmick Aug 21 '16 at 14:28