Complex Eigenvalues

Complex Properties

• $\overline{(x+y)}=\overline{x}+\overline{y}$
• $\overline{A\vec{v}}=A\overline{\vec{v}}$
• $\text{Lm}(x\overline{x})=0$

Complex Roots of the Characteristic Polynomial

All real numbers are complex numbers so, Every polynomial of degree $n$ has exactly $n$ complex roots, counting multiplicity.

If $\lambda$ is an eigenvalue of real matrix $A$ with eigenvector $\vec{v}$, then $\overline{\lambda }$ is an eigenvalue of $A$ with eigenvector $\overline{\vec{\lambda }}$.