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Particular types of matrices
Definition of eigenvalue of a matrix
Definition of eigenvector of a matrix
Eigenspaces
Every matrix has an eigenvalue over the complex numbers.
Eigenvalues and operations on matrices
Eigenvectors with distinct eigenvalues are linearly independent.
Multiplicity
Characteristic and minimal polynomials
The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.
Definition of eigenvalue/characteristic value of a linear transformation
Definition of eigenvector/characteristic vector of a linear transformation
Definition of characteristic polynomial of a linear transformation
Definition of minimal polynomial of a linear transformation
The Cayley-Hamilton theorem for a linear transformation
The minimal polynomial of a linear transformation exists and is unique.
Definition of applying a polynomial to a linear transformation
A linear transformation on a finite dimentional nontrivial vector space has at least one eigenvalue.
Definition of eigenspace of a linear transformation
The eigenspace of a linear transformation is a nontrivial subspace.
Definition of invariant subspace of a linear transformation.
If a space is the direct sum of invariant subspaces

The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.
The eigenvalues of a power of a matrix are the power the eigenvalues.
The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.
The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.
A matrix and its transpose have the same eigenvalues/characteristic polynomial.

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