lecture_37

# lecture_37 - MA 36600 LECTURE NOTES FRIDAY APRIL 23...

This preview shows pages 1–2. Sign up to view the full content.

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

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

Unformatted text preview: MA 36600 LECTURE NOTES: FRIDAY, APRIL 23 Diagonalizing 2 × 2 Matrices Distinct Eigenvalues. We explain how we chose the matrices D and T in the previous example. Indeed, we discuss a general theory which holds especially well for 2 × 2 matrices. Consider such a matrix: A = a 11 a 12 a 21 a 22 . Recall that we can define the trace , determinant , and discriminant : tr A = a 11 + a 22 , det A = a 11 a 22- a 12 a 21 ; disc A = (tr A ) 2- 4(det A ) = ( a 11- a 22 ) 2 + 4 a 12 a 21 . We assume that disc A 6 = 0. Then p A ( r ) = r 2- (tr A ) r + (det A ) has two distinct roots r 1 and r 2 . First we give formulas for the eigenvectors of such a matrix when the eigenvalues are known. Observe that if we choose ξ = a 12 r- a 11 = ⇒ ( r I- A ) ξ = r- a 11- a 12- a 21 r- a 22 a 12 r- a 11 = p A ( r ) . In particular, when r = r 1 the right-hand side is the zero vector. This means we have eigenvectors ξ (1) = a 12 r 1- a 11 and ξ (2) = a 12 r 2- a 11 ....
View Full Document

## This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue.

### Page1 / 3

lecture_37 - MA 36600 LECTURE NOTES FRIDAY APRIL 23...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online