Eigenvalues & Eigenvectors
Definitions
Characteristic Polynomial
Let
Eigenvalues
Roots of the equation
Eigenvectors
The column vectors satisfying the equation
Normalized eigenvectors
An eigenvector with the magnitude (norm) of
Norm
Norm of a column or row matrix
Algebraic Multiplicity
If the characteristic polynomial consists of a factor of the form
Spectrum
Set of all eigenvalues.
Spectral Radius
Linear Independence of Eigenvectors
Suppose
All those eigenvectors are independent iff:
For special matrices
Real symmetric matrix
Suppose
- The eigenvalues of
are all real: - The eigenvectors of
(corresponding to distinct values of ) are mutually orthogonal
Upper triangular matrix
The eigenvalues are the diagonal entries.