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3 Eigenvalue Problem
Problem 3.5 Find the eigenvalues and the eigenvectors of the (3, 3)-matrix
a =
⎡
⎣
1 1 2
1 1 3
2 3 2
⎤
⎦ ,
plot the eigenvectors and show that they are orthogonal in the sense of Eq. 3.4.
Problem 3.6 Find the eigenvalues and the right/left eigenvectors of the (2, 2)matrix
a =
1 5
0 3
by hand calculations and MATLAB. Show that the eigenvectors are orthogonal
sense of Eq. 3.4.
Problem 3.7 Consider a vector x in R 3 with components (1, −2, 3) in the standard
reference {i k }, k = 1, 2, 3. Find the components of x in the coordinates defined by
the eigenvectors {x k }, k = 1, 2, 3, of the matrix in Problem 3.5. Plot the eigenvectors
{x k } and the vector x in the reference {i k }. Plot also the unit vectors {i k } and the
vector x in the reference defined by the eigenvectors {x k }.
Reference
1. P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd edn. (Academic Press Inc., New
York, 1985)