Examples on Eigenvalues and Eigenvectors

Comp.145  November 1996  Dr. A.J. Mestel

1. Find the eigenvalues and normalized eigenvectors of the symmetric matrix

( \begin{matrix}4 & −2 & 0 \\ −2 & 3 & −2 \\ 0 & −2 & 2 \end{matrix} ) .

Verify that the sum of the eigenvalues and the dot products of the eigenvectors with each other are as you would expect.

2. Suppose detA ≠ 0. How do the eigenvalues and eigenvectors of A^{−1} relate to those of A?

3. The eigenvalues λ_{i} of a matrix A satisfy the characteristic equation

λ^{n} + b_{n−1}λ^{n−1} + … + b_{1}λ + b_{0} = 0 .

The constants b_{i} can be used to define another matrix, B, by

B = A^{n} + b_{n−1}A^{n−1} + … + b_{1}A + b_{0}I .

Show that B\underline{e}_{i} = 0, where \underline{e}_{i} is any eigenvector of A. Deduce that B\underline{x} = 0 where \underline{x} is any vector expressable in terms of the eigenvectors, i.e. \underline{x} = ∑ α_{i}\underline{e}_{i} for any constants α_{i}. [It follows from this, that provided the eigenvectors \underline{e}_{i} are linearly independent, then B = 0. That is, the matrix A satisfies its own characteristic equation. This is a simplified version of the Cayley-Hamilton theorem.]

4. Many iterative algorithms are equivalent to multiplying a vector \underline{x} by a matrix A each iteration, so that if the initial state is \underline{x} = \underline{x}_{0}, after m iterations \underline{x} = A^{m}\underline{x}_{0}. Consider the simple system described by the 3 × 3 matrix given below along with its eigenvalues and (non-normalized) eigenvectors:

A = \frac{1}{4} ( \begin{matrix}4 & 1 & 1 \\ 1 & 3 & 2 \\ 1 & 2 & 3 \end{matrix} )  e-vals: \{ \begin{matrix}λ_{1} & = 1.5 \\ λ_{2} & = 0.25 \\ λ_{3} & = 0.75 \end{matrix}  e-vecs: \{ \begin{aligned}\underline{e}_{1} & = (1, 1, 1) \\ \underline{e}_{2} & = (0, 1, −1) \\ \underline{e}_{3} & = (2, −1, −1) \end{aligned} .

What would you expect to happen as m → ∞ if \underline{x}_{0} = 2\underline{e}_{2} + \frac{1}{10} \underline{e}_{3} = (0.2, 1.9, −2.1)? Write a program to calculate 100 iterations of the process, starting with this value. You will probably find that something quite different from what you predict occurs. Can you explain this?