Let A = 2 1 1 1 2 1 1 1 2 (a) Find the characteristic...

Let A = 2 1 1

1 2 1

1 1 2

(a) Find the characteristic polynomial PA(λ) of A and the eigenvalues of A. For convenience, as usual, enumerate the eigenvalues in decreasing order λ1 ≥ λ2 ≥ λ3.

(b) For each eigenvalue λ of A find a basis of the corresponding eigenspace V (λ). Determine (with a motivation) whether V (λ) is a line or a plane through the origin. If some of the spaces V (λ) is a plane find an equation of this plane.

(c) Find a basis of R 3 consisting of eigenvectors if such basis exist. (Explain why or why not). Is the matrix A diagonalizable? If ”yes”, then write down a diagonalizing matrix P, and a diagonal matrix Λ such that A = PΛP −1 , P −1AP = Λ. Explain why the matrix P is invertible but do not compute P −1 .

(d) Consider the eigenvalues λ1 > λ3. Is it true that the orthogonal complements of the eigenspaces satisfy (Vλ1 ) ⊥ = Vλ3 , (Vλ3 ) ⊥ = Vλ1 ? Why or why not??

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Expert Solution

Colby Messinger answered 1 year ago

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