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

Result R5079 on D5862: Complex matrix eigenvalue
Subresult of R5081: Complex matrix determinant zero iff some nonzero vector is mapped to zero

Characterisation of complex matrix eigenvalues in terms of characteristic polynomial

Formulation 0

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
Let $\lambda \in \mathbb{C}$ be a D1207: Complex number.

Then the following statements are equivalent

(1)	\begin{equation} \exists \, z \in \mathbb{C}^{N \times 1} \setminus \{ \boldsymbol{0} \} : A z = \lambda z \end{equation}
(2)	\begin{equation} \text{Det}(A - \lambda I_N) = 0 \end{equation}

Proofs

Proof 0

Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
Let $\lambda \in \mathbb{C}$ be a D1207: Complex number.

Since we can write the first statement in the equivalent form \begin{equation} \exists \, z \in \mathbb{C}^{N \times 1} \setminus \{ \boldsymbol{0} \} : (A - \lambda I_N) z = \boldsymbol{0} \end{equation} then this result is a subresult to R5081: Complex matrix determinant zero iff some nonzero vector is mapped to zero. $\square$