Math416_SchurDecomposition - Math 416 - Abstract Linear...

Info iconThis preview shows pages 1–2. 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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Schur decomposition Let us illustrate the algorithm to find a Schur decomposition, as in 6.1, Theorem 1.1. Example: Find a Schur decomposition of the matrix A = 7- 2 12- 3 . Solution: First, we want an eigenvector of A . Let us find the eigenvalues: det( A- I ) = 7- - 2 12- 3- = (7- )(- 3- ) + 24 = 2- 4 - 21 + 24 = 2- 4 + 3 = ( - 1)( - 3) . The eigenvalues are = 1 , 3. We could arbitrarily pick one of the two and find an eigenvector, but while were at it, lets find both: = 1 : A- I = A- I = 6- 2 12- 4 3- 1 Take 1 3 , normalized to 1 10 1 3 . = 3 : A- I = A- 3 I = 4- 2 12- 6 2- 1 Take 1 2 , normalized to 1 5 1 2 . In fact, lets pick 1 = 3 with normalized eigenvector u 1 = 1 5 1 2 . We need to find an orthonormal basis { v 2 } of Span { u 1 } = Span { 1 2 } = Span {- 2 1 } . Pick v 2 = 1 5- 2 1 ....
View Full Document

This note was uploaded on 12/16/2011 for the course MATH 416 taught by Professor Frankland during the Fall '11 term at University of Illinois at Urbana–Champaign.

Page1 / 4

Math416_SchurDecomposition - Math 416 - Abstract Linear...

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

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