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Lecture 31 - Mar 23

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

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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

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

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