Eigenvalues and Eigenvectors (11/17/04) We know that every linear transformation T fixes the 0 vector (in R n, fixes the origin). But are there other subspaces in the domain of T which are left fixed by T ‘s action? If so, the vectors in those subspaces are called eigenvectors. “Eigen” is German for “own.”

A familiar example In our Markov Chain lab, the transition matrices A had steady-state vectors x, i.e., A x = x. We were interested in the particular vector x which was a probability vector, but the whole line spanned by x is also fixed. In this case A has the same effect on x as multiplication by 1, so 1 is called an eigenvalue of A.

Definitions An eigenvector of an n by n matrix A is a nonzero vector x such that A x = x for some scalar (“lambda”). A scalar is called an eigenvalue of A if there exists a vector x such that A x = x. The eigenspace of A associated with an eigenvalue is the 0 vector and the set of all of ’s eigenvectors. (It is a subspace. Check!)

Examples Show that x = (6,-5) is an eigenvector of A =. What is its eigenvalue? 2 is an eigenvalue for A =. Find a basis for its eigenspace.

Theorems on eigenvalues Theorem. is an eigenvalue of a matrix A if and only if the homogeneous equation (A - I) x = 0 has nontrivial solutions. Theorem. The eigenvalues of an upper (or lower) triangular matrix are simply the entries along the diagonal. Corollary. The determinant of any matrix is plus or minus the product of its eigenvalues.

Assignment for Friday Begin work on Hand-in #3. Read Section 5.1. Do Exercises 1 – 9 odd, 15 – 25 odd.