HW6 - x ∈ V we have M x ∈ V d Show that V j is an...

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Math 601 HOMEWORK 6 Solve the problems 1. , 2. below, and also from Strang, 3rd ed., problems 5.2.1, 5.2.3, 5.2.5, 5.2.6, 5.2.7 on p. 260-261 (see the .pdf file handed out in connection to HW 5). 1. Consider an n × n matrix M with entries in F ( = R or C ) with eigenvalues λ 1 ,...,λ n in F . Definition 1. The set V λ j = { x F n | M x = λ j x } is called the eigenspace of M associated to the eigenvalue λ j . a) Show that V λ j is the null space of the transformation M - λ j I . b) Show that V λ j is a subspace of F n . c) Are all the vectors in V λ j eigenvectors of M ? Definition 2. A subspace V is called an an invariant subspace for M if M ( V ) V (which means that for any
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Unformatted text preview: x ∈ V we have M x ∈ V ). d) Show that V λ j is an invariant subspace for M . e) Show that V λ j ∩ V λ k = { } if λ j 6 = λ k . f) Denote by λ 1 ,...,λ k the distinct eigenvalues of M (in other words, { λ 1 ,...,λ k } = { λ 1 ,...,λ n } ). Show that M is diagonalizable in F n if and only if V λ 1 ⊕ ... ⊕ V λ k = F n and explain why the sum before is a direct sum. 2. Find the eigenspaces of the matrix A = ± 2 1 0 2 ² Is the matrix diagonalizable? 1...
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