eigenvalues - CANONICAL FORMS OF 2 2 MATRICES AND THEIR...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

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

Unformatted text preview: CANONICAL FORMS OF 2 2 MATRICES AND THEIR APPLICATIONS KIAM HEONG KWA 1. Canonical Forms of 2 2 Matrices The characteristic equation of a 2 2 complex matrix, say A = a 11 a 12 a 21 a 22 , is the quadratic equation det( A- I 2 ) = 2- (tr A ) + det A = 0 , where I 2 = 1 0 0 1 is the 2 2 identity matrix and tr A = a 11 + a 22 and det A = a 11 a 22- a 12 a 21 are respectively the trace and the determinant of A . Thus A has two distinct eigenvalues given by 1 , 2 = tr A p (tr A ) 2- 4 det A 2 , except when tr A = 2 det A so that 1 = 2 . The goals of this section are (1) to show that there is an invertible 2 2 complex matrix S such that A can be Jordan decomposed into one of the following forms: S 1 2 S- 1 , S 1 1 S- 1 = 1 I 2 , S 1 1 1 S- 1 ; (2) to identify such a matrix S ; (3) to give a sufficient and necessary condition on the eigenvalues of A so that A has a particular Jordan decomposition. In either of the Jordan decomposition A = SJS- 1 , the matrix J is called a Jordan canonical form of A . Date : August 17, 2010. 1 2 KIAM HEONG KWA To begin with, let us consider the case 1 6 = 2 . Let s i = s 1 i s 2 i be an eigenvector associated with i , i = 1 , 2. Since 1 6 = 2 , the vectors s 1 and s 2 are linearly independent. Thus the matrix S = ( s 1 s 2 ) = s 11 s 12 s 21 s 22 is invertible. Now AS = a 11 a 12 a 21 a 22 s 11 s 12 s 21 s 22 = a 11 s 11 + a 12 s 21 a 11 s 12 + a 12 s 22 a 21 s 11 + a 22 s 21 a 21 s 12 + a 22 s 22 = ( A s 1 A s 2 ) = ( 1 s 1 2 s 2 ) = 1 s 11 2 s 12 1 s 21 2 s 22 = s 11 s 12 s 21 s 22 1 2 = S 1 2 , from which it follows that A = S 1 2 S- 1 . Next, let us consider the case 1 = 2 and there are two linearly independent eigenvectors s 1 and s 2 associated with 1 . By setting S = ( s 1 s 2 ) , it can be shown as in the previous case that A = S 1 1 S- 1 = I 2 thanks to the linear independence of s 1 and s 2 . This implies that A has only one eigenvalue with two linearly inde- pendent eigenvectors if and only if A is a scalar matrix, i.e., a multiple of I 2 . Finally, let us consider the case 1 = 2 and there is only one eigenvector t 1 = t 11 t 21 associated with 1 up to linear independence. From the preceding paragraph, we know that A is not a scalar ma- trix. We first show that there is an invertible matrix T such that A = T 1 1 T- 1 for some scalar 6 = 0. Replacing t 1 by t 1 k t 1 k if necessary, we may assume that t 1 is a unit vector. Let t 2 = t 12 t 22 be CANONICAL FORMS OF 2 2 MATRICES AND THEIR APPLICATIONS 3 a unit vector in C 2 that is orthogonal to t 1 so that t 11 t 12 + t 21 t 22 = 0....
View Full Document

This note was uploaded on 11/11/2011 for the course MATH 415.01 taught by Professor Kwa during the Winter '11 term at Ohio State.

Page1 / 9

eigenvalues - CANONICAL FORMS OF 2 2 MATRICES AND THEIR...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online