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

Result R5081 on D5717: Complex matrix determinant

Complex matrix determinant zero iff some nonzero vector is mapped to zero

Formulation 0

Let $A \in \mathbb{C}^{N \times N}$ be a D999: Complex matrix.

Then the following statements are equivalent

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

Subresults

▶	R5079: Characterisation of complex matrix eigenvalues in terms of characteristic polynomial

Proofs

Proof 0

Let $A \in \mathbb{C}^{N \times N}$ be a D999: Complex matrix.