Applied Numerical Methods Using MATLAB, 2nd Edition

8Matrices and Eigenvalues

Chapter Outline

• 8.1 Eigenvalues and Eigenvectors
• 8.2 Similarity Transformation and Diagonalization
• 8.3 Power Method
  • 8.3.1 Scaled Power Method
  • 8.3.2 Inverse Power Method
  • 8.3.3 Shifted Inverse Power Method
• 8.4 Jacobi Method
• 8.5 Gram‐Schmidt Orthonormalization and QR Decomposition
• 8.6 Physical Meaning of Eigenvalues/Eigenvectors
• 8.7 Differential Equations with Eigenvectors
• 8.8 DoA Estimation with Eigenvectors^[Y-3]
• Problems

In this chapter, we will look at the eigenvalue or characteristic value λ and its corresponding eigenvector or characteristic vector v of a matrix.

8.1 Eigenvalues and Eigenvectors

The eigenvalue or characteristic value and its corresponding eigenvector or characteristic vector of an N × N matrix A are defined as a scalar λ and a nonzero vector v satisfying

(8.1.1)

where (λ, v) is called an eigenpair, and there are N eigenpairs for the N × N matrix A.

How do we get them? Noting that

• in order for the aforementioned equation to hold for any nonzero vector v, the matrix [A − λI] should be singular, i.e. its determinant should be zero (|A − λI| = 0), and
• the determinant of the matrix [A − λI] is a polynomial of degree N in terms of λ,

we first must find the eigenvalue λ_i's by solving the so‐called characteristic equation

(8.1.2)

and then substitute the λ_i's, one by one, into Eq. (8.1.1) to solve it for the eigenvector v_i's. ...