Eigenvalues and eigenvectors
Eigenvalues and eigenvectors capture the intrinsic stretching/compressing directions of a linear transformation. For a square matrix $A$, a nonzero vector $v$ is an eigenvector with eigenvalue $\lambda$ if $Av = \lambda v$. Eigenvalues reveal fundamental properties: invertibility, stability, diagonalizability, and more.
Parents
Children
- 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