7.1 Eigenvalues and Eigenvectors Hwk: (1 to14)b, 15a, 20. Also see 23, but don't do for homework
See: Eigenvector_Calculator 1, 2, 3, 4, for various versions of on-line Eigenvalue/vector calculators. If you would like to see more, try a Google search for eigenvector calculator.
Def of eigenvector: one for which Ax = λx That is, vector x for which mult of x by A gives a result parallel to x.
Characteristic eqn/polynomial: det (λI – A) = 0
If A has dimension n by n, this is a polynomial of degree n,
so there are n eigenvalues, some of which may be repeated.
Text mentions the practical difficulty of finding all the solutions of a polynomial equation in n unknowns
Easy case: Thm 1: Triangular matrix: eigenvalues are simply the entries on the main diagonal.
For an example, review the solution for (See Thm 2)
eigenvalues λ
eigenvectors (solutions of the homogeneous system (λI – A)x = 0 for each λ
There will always be free variables, so we will always get parameterized (nontrivial) solutions
So the eigenvectors will be bases of the nullspace of the matrices (λI – A)
Thm 3: Powers of matrices: If λ is an eigenvalue of A, then λk is an eigenvector of Ak
Thm 4: A square matrix is invertible iff λ = 0 is not an eigenvalue of A
Illustrate with a diagonal matrix:
If it has a 0 on the main diagonal, it's not invertible, and 0 is an eigenvalue.
7.2 Diagonalization Hwk: 1, 2, 5, 11, 13, 19, 20c, 22, 25
7.3 Orthogonal Diagonalization Hwk: 1abc, 2, 7, 15
Application of this lesson: do lesson 9.6: applying quadratic forms (described in 9.5) to eliminate the cross-product term from rotated conics, to analyze the resulting conic section
Return to: Merced College; Don Power Updated 05/08/07 by Don Power