Eigenvalues and Eigenvectors

This chapter systematically studies the eigenvalues and eigenvectors of matrices, similarity, diagonalization, and properties of real symmetric matrices.

Definition of Eigenvalues and Eigenvectors

$A\vec{x} = \lambda \vec{x}$
$|A - \lambda I| = 0$ is the characteristic equation

Properties and Computation

An $n$ -order matrix has $n$ eigenvalues (counting multiplicities)
Eigenvectors corresponding to different eigenvalues are linearly independent
Eigenvectors corresponding to different eigenvalues are linearly independent

Similarity and Diagonalization

Similarity: $B = P^{-1}AP$
Diagonalizability condition: there are $n$ linearly independent eigenvectors
Real symmetric matrices can be orthogonally diagonalized

Exercises

Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ .
Determine whether $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is diagonalizable.
If $A$ has eigenvalues $1,2,3$ , write $|A|$ and $\operatorname{tr}A$ .
Determine whether $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is a real symmetric matrix, and whether all its eigenvalues are real.
If $A$ is diagonalizable, write its diagonalization form.

Reference Answers

1. Eigenvalues and eigenvectors

$|A - \lambda I| = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = 0$ , $\lambda=1,3$

For $\lambda=1$ , $\vec{x} = (1,-1)$ ; for $\lambda=3$ , $\vec{x} = (1,1)$

2. Diagonalizability

There is only one eigenvector, so it is not diagonalizable.

3. $|A|$ and $\operatorname{tr}A$

$|A| = 1\times2\times3 = 6$ , $\operatorname{tr}A = 1+2+3=6$

4. Real symmetry and eigenvalues

$A$ is a real symmetric matrix, eigenvalues are $1,-1$ , both real.

5. Diagonalization form

$A = PDP^{-1}$ , $D$ is a diagonal matrix, $P$ is the matrix of eigenvectors.