MATA23H3 Lecture Notes - Lecture 24: Diagonalizable Matrix, Diagonal Matrix, Invertible Matrix
Document Summary
Mata23 - lecture 24 - similar matrices and cayley-hamilton theorem. Similar matrices: if an n n matrix a is similar to a diagonal matrix d, i. e. a = p 1dp where p is an n n invertible matrix, then ak = p 1dkp. A2 = aa = p 1dp p 1dp (note: p p 1 = i) P 1 k dkpk: example 8: find a100, where a = 2 0 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) = (2 )( 2 1) = (2 )( 1)( + 1) 1 = 1, 2 = 1, 3 = 2. Let p = (cid:126)v1 (cid:126)v2 (cid:126)v3. A = p dp 1, ak = p dkp 1. Cayley-hamilton theorem: every square matrix satis es its characteristic equation. P ( ) = det(a i) = 0. Let det(a i) = c0 n + c1 n 1 + c2 n 2 + + cn 1 + cn = 0.
