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
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 = a 12 r a 11 = ( r I A ) = 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 righthand side is the zero vector. This means we have eigenvectors (1) = a 12 r 1 a 11 and (2) = a 12 r 2 a 11 ....
View Full
Document
 Spring '09
 MA
 Matrices

Click to edit the document details