Let A be an n X n matrix with n real eigenvalues, counting multiplicities, denoted by λ1,,,,,,, λn. It can be shown that A admits a (real) Schur factorization. Parts (a) and (b) show the key ideas in the proof. The rest of the proof amounts to repeating (a) and (b) for successively smaller matrices, and then piecing together the results.
a. Let u1 be a unit eigenvector corresponding to λ1, let u2,,,,,,,,,, un be any other vectors such that {u1,,,,,,,, un} is an orthonormal basis for Rn, and then let U = [u1 u2 ......un]. Show that the first column of U T AU is λ1e1, where e1 is the first column of the n n identity matrix.
b. Part (a) implies that U TAU has the form shown below. Explain why the eigenvalues of A1 are λ2,,,,,,,,,,, λn
When the right side of an equation Ax = b is changed slightly-say, to Ax = b +?b for some vector ?b-the solution changes from x to x + ?x, where ?x satisfies A (?x) = ?b. The quotient ||?b||/||b|| is called the relative change in b (or the relative error in b when ?b represents possible error in the entries of b). The relative change in the solution is ||x||/||x||. When A is invertible, the condition number of A, written as cond (A), produces a bound on how large the relative change in x can be:
