Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Enclosed binary operation
Groupoid
Ringoid
Semiring
Ring
Left ring action
Module
Linear combination
Linear map
Eigenvector
Eigenvalue
Complex matrix eigenvalue
Complex matrix eigenvalue sequence
Formulation 0
Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
Let $I_N \in \mathbb{C}^{N \times N}$ be a D5699: Complex identity matrix.
A D5975: Euclidean complex number $(\lambda_1, \ldots, \lambda_N) \in \mathbb{C}^N$ is an eigenvalue sequence for $A$ if and only if \begin{equation} \forall \, z \in \mathbb{C} : \text{Det}(z I_N - A) = \prod_{n = 1}^N (z - \lambda_n) \end{equation}
Results
» R5531: Eigenvalue sequence exists for every complex square matrix
» R5535: Reciprocals form an eigenvalue sequence for complex matrix inverse
» R5536: Eigenvalues of a conjugate symmetric complex matrix are real
» R5564: Eigenvalue sequence for an upper triangular complex matrix
» R5565: Eigenvalue sequence for a lower triangular complex matrix
» R5566: Eigenvalue sequence for a diagonal complex matrix
» R5567: Eigenvalue sequence for a diagonal complex matrix with constant diagonal
» R5568: Eigenvalue sequence for an identity complex matrix
» R5563: Eigenvalue sequence for a triangular complex matrix