Thus q is diagonal and unitary and d11 1 0 0 d22

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . 0 0 ... 0 dnn ωn Since the matrix D∞ Q∞ commutes with A∞ , and A∞ is unitarily equivalent to A (follows from the QR algorithm), D∞ Q∞ is unitarily equivalent to A. So λj = djj ωj are the eigenvalues of A. Thus, we know |djj | = |λj |. Moreover, if the diagonal entries of D∞ are distinct (magnitudes of the eigenvalues of A are distinct), and since D∞ Q∞ = Q∞ D∞ , the off diagonal entries of Q∞ ￿ must be zeros, and we precisely have Q∞ = Q∞ . Thus, Q∞ is diagonal and unitary, and d11 ω1 0 ... 0 . . . d22 ω2 . . . 0 A∞ = . . ... ... . . 0 0 ... 0 dnn ωn 2 So if the magnitudes of a normal matrix A are distinct and if the QR algorithm converges, then it diagonalizes A. 1 Real Matrices (cont’d) We proceed to show under which conditions a matrix with entries over R can be diagonalized the same way as matrices with complex entries. 1.5.1 Definition. A matrix A ∈ Mn (R) is similar to B ∈ Mn (R) if there exists invertible S ∈ Mn (R) such that B = S −1 AS. The matrix A is diagonalizable if A is similar to a diagonal matrix. 1.5.2 Theorem. A ∈ Mn (R) is diagonalizable if and only if there is a set of n linearly independent eigenvectors. Proof. As before, S −1 AS = D, with D diagonal implies that S = [x1 , x2 , · · · , xn ] contains a basis of n eigenvectors and vice versa. 1.5.3 Theo...
View Full Document

This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

Ask a homework question - tutors are online