Eigenvalues & Eigenvectors

Definitions

Characteristic Polynomial

Let be a matrix.

Eigenvalues

Roots of the equation are the eigenvalues of .

Note

• Product of the eigenvalues is equal to the determinant of the matrix
• Sum of the eigenvalues is equal to the trace of a matrix
• If is an eigenvalue of , then is an eigenvalue of
• and have the same eigenvalues.

Eigenvectors

The column vectors satisfying the equation .

Normalized eigenvectors

An eigenvector with the magnitude (norm) of . Normalizing factor of any eigenvector is:

Norm

Norm of a column or row matrix is denoted by and defined as:

Algebraic Multiplicity

If the characteristic polynomial consists of a factor of the form and is not a factor of the characteristic polynomial then is the algebraic multiplicity of the eigenvalue .

Spectrum

Set of all eigenvalues.

Spectral Radius

Linear Independence of Eigenvectors

Suppose is a set of eigenvectors. is a set of scalars.

All those eigenvectors are independent iff:

For special matrices

Real symmetric matrix

Suppose is a symmetric matrix with all real entries. Then:

• The eigenvalues of are all real:
• The eigenvectors of (corresponding to distinct values of ) are mutually orthogonal
• and have the same eigenvalues

Upper triangular matrix

The eigenvalues are the diagonal entries.