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Unformatted text preview: Wednesday March 25 Lecture 32 : More on eigenvectors. (Refers to 6.2 and 6.3) Expectations: 1. Define an eigenspace. 2. Recognize that 1 is an eigenvalue of A iff 1/ 1 is an eigenvalue of A-1 . 3. Define algebraic and geometric multiplicity of an eigenvalue. 32.1 Definition Let 1 be an eigenvalue of the matrix A . We have seen that we can associate to 1 a set of non-zero vectors E 1 called eigenvectors of A associated to 1 . This set is the complete solution of the homogeneous system ( A 1 I ) x = , minus the zero vector . If we include the zero vector in this set we recognize that it is a subspace of R n . When the zero vector is included we call E 1 the eigenspace associated to 1 . 32.1.1 Remark Given a matrix A we can associate to each of its eigenvalues an eigenspace E , 32.2 Proposition Let A be an invertible matrix. If 1 is an eigenvalue of A then 1/ 1 is an eigenvalue of A-1 . Furthermore x is an eigenvector associated to 1/ 1 iff x is an eigenvector of 1 . Proof: Suppose matrix A .is invertible. 1 is a eigenvalue of A ....
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