Lecture 31 - Mar 23

Lecture 31 - Mar 23 - Monday March 23 Lecture 31 :...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Monday March 23 Lecture 31 : Eigenvectors associated to an eigenvalue. (Refers to 6.2) Expectations: 1. Define the family of all eigenvectors associated to an eigenvalue. 2. Find all eigenvectors associated to an eigenvalue. 3. Define an eigenspace. 4. Recognize that λ 1 is an eigenvalue of A iff 1/ λ 1 is an eigenvalue of A -1 . 5. Define algebraic and geometric multiplicity of an eigenvalue. 31.1 Theorem Let λ 1 be a number. The number λ 1 is an eigenvalue of A iff the the system A x = λ 1 x has a non-trivial solution. Proof : The number λ 1 is an eigenvalue of A is a solution to the characteristic equation det( A λ I ) = 0 det( A λ 1 I ) = 0 the matrix A λ 1 I is not invertible ( A λ 1 I ) x = 0 has a nontrivial solution (infinitely many solutions) A x = λ 1 x has a non-trivial solution as required. 31.1.1 Example Show that 1 is an eigenvalue of the following matrix A 2 0 0 0 3 1 1 0 1 Solution: By the above theorem we need only verify that the following homogeneous system has a non-trivial solution
Background image of page 1

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

View Full DocumentRight Arrow Icon
2 – 1 0 0 0 3 –1 1 x = 0 1 0 1 – 1 This coefficient matrix easily row-reduces to 1 0 0 0 2 1 0 0 0 Since ( A – (1) I ) x = 0 has a non-trivial solution then 1 is an eigenvalue of A . So if λ 1 is an eigenvalue of A then there exists infinitely many vectors x such that A x = λ 1 x . We have a special name for vectors x which are associated to an eigenvalue λ 1 in this way, i.e., non-zero vectors x such that A x = λ 1 x . We give it in the following definition.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 06/10/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

Page1 / 6

Lecture 31 - Mar 23 - Monday March 23 Lecture 31 :...

This preview shows document pages 1 - 3. Sign up to view the full document.

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