This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2 Lecture Notes Fall 2007 Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 Chapter 2 2.3: Diagonalization and Eigenvalues Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.3: Diagonalization and Eigenvalues Powers of a Matrix There are many possible ways to introduce eigenvalues and eigenvectors, but one computational approach is to consider powers of a square matrix. Consider the matrix A = 3 1 3 5 . You can check that A 2 = 3 1 3 5 3 1 3 5 = 12 8 24 28 and A 3 = A 2 A = 12 8 24 28 3 1 3 5 = 60 52 156 164 . In general, it is very cumbersome to calculate higher powers of a matrix! Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.3: Diagonalization and Eigenvalues Diagonal Matrices A diagonal matrix D is a square matrix such that all entries off its main diagonal are zero. For example, D = 6 0 0 2 . It is easy to check that D 2 = 36 0 4 and D 3 = 216 0 8 and that in general, D n = 6 n 2 n Even the inverse of D is easy to find: D 1 = 1 / 6 1 / 2 . Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.3: Diagonalization and Eigenvalues Properties of Diagonal Matrices Let D be the n × n diagonal matrix with elements d 11 , d 22 , . . . , d nn on its main diagonal. Nicholson writes this matrix as D = diag( d 11 , d 22 , . . . , d nn ) . Then D k = diag d k 11 , d k 22 , . . . , d k nn . Also, det D = d 11 d 22 ··· d nn . Thus D is invertible if and only if all of its diagonal elements are nonzero, and then D 1 = diag(1 / d 11 , 1 / d 22 , . . . , 1 / d nn ) . Wouldn’t it be nice if all matrices were diagonal! Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.3: Diagonalization and Eigenvalues Similar Matrices We say that two n × n matrices A and B are similar if there is an invertible n × n matrix P , such that A = P 1 BP . Similar matrices have, as the name suggests, many things in common. For instance: det A = det( P 1 BP ) = det P 1 det B det P = (det P ) 1 det B det P = det B . Similar matrices have the same rank, to mention one other property, that we are familiar with. Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.3: Diagonalization and Eigenvalues Powers of P 1 BP ( P 1 BP ) 2 = ( P 1 BP )( P 1 BP ) = P 1 ( B ( PP 1 ) B ) P = P 1 ( BIB ) P = P 1 B 2 P Similarly, ( P 1 BP ) k = P 1 B k P , for k > , as you can check. Thus if A and B are similar, A k = P 1 B k P . Note: if Q = P 1 , then A = P 1 BP ⇔ A = QBQ 1 , so the “side” that the inverse matrix is on, is a matter of choice. Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.3: Diagonalization and Eigenvalues Example 1 Let A = 3 1 3 5 and consider the matrix P = 1 1 3 1 ....
View
Full
Document
This note was uploaded on 06/22/2008 for the course ENGINEERIN MAT188 taught by Professor Burbula during the Fall '07 term at University of Toronto.
 Fall '07
 Burbula

Click to edit the document details