ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5556 on D5862: Complex matrix eigenvalue

Squared eigenvalue is an eigenvalue for the square of a complex matrix

Formulation 0

Let $A \in \mathbb{C}^{N \times N}$ be a D5862: Complex matrix eigenvalue such that

(i)	$\lambda \in \mathbb{C}$ is a D5862: Complex matrix eigenvalue for $A$

Then $\lambda^2$ is a D5862: Complex matrix eigenvalue for $A^2$.

Proofs

Proof 0

Let $A \in \mathbb{C}^{N \times N}$ be a D5862: Complex matrix eigenvalue such that

(i)	$\lambda \in \mathbb{C}$ is a D5862: Complex matrix eigenvalue for $A$

Let $z \in \mathbb{C} \setminus \{ \boldsymbol{0} \}$ be a nonzero complex number. Then \begin{equation} \begin{split} A^2 z = A (A z) = A \lambda z = \lambda A z = \lambda^2 z \end{split} \end{equation} $\square$