This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 7.4 Diagonalization Fact 7.4.1 The matrix of a linear trans- formation with respect to an eigenbasis is diagonal Consider a transformation T~x = A~x , where A is an n n matrix. Suppose B is an eigenba- sis for T consisting of vectors ~v 1 ,~v 2 ,...,~v n , with A~v i = i ~v i . Then the B-matrix D of T is D = S- 1 AS = 1 . 2 . . . . . 0 0 n Here S = | | | ~v 1 ~v 2 ... ~v n | | | ~x A- A~x S S h ~x i S- D h A~x i S ~x = S h ~x i S h ~x i S = S- 1 ~x 1 Def 7.4.2 Diagonalizable matrices An n n matrix A is called diagonalizable if A is similar to a diagonal matrix D , that is, if there is an invertible n n matrix S such that D = S- 1 AS is diagonal. Fact 7.4.3 Matrix A is diagonalizable iff there is an eigen- basis for A . In particular, if an n n matrix A has n distinct eigenvalues, then A is diagonal- izable....
View Full Document
This note was uploaded on 03/01/2012 for the course MATH 100 taught by Professor Evans during the Spring '12 term at Seneca.
- Spring '12