Let A be a 4 4 real symmetric matrix with eigenvalues 1 1, 2 3 4 0 (a) Explain why the multiple eigenvalue 0 must have three linearly independent eigenvectors x2, x3, x4 (b) Let x1 be an eigenvector belonging to 1 How is x1 related to x2, x3, x4 Explain (c) Explain how to use x1, x2, x3, x4 to construct an orthogonal matrix U that diagonalizes A (d) What type of matrix is eA Is it symmetric Is it positive definite Explain your answers

Question: Let A be a 4 4 real symmetric matrix with eigenvalues 1 = 1, 2 = 3 = 4 = 0 (a) Explain why

Let A be a 4 × 4 real symmetric matrix with eigenvalues
λ1 = 1, λ2 = λ3 = λ4 = 0
(a) Explain why the multiple eigenvalue λ = 0 must have three linearly independent eigenvectors x2, x3, x4.
(b) Let x1 be an eigenvector belonging to λ1. How is x1 related to x2, x3, x4? Explain.
(c) Explain how to use x1, x2, x3, x4 to construct an orthogonal matrix U that diagonalizes A.
(d) What type of matrix is eA? Is it symmetric? Is it positive definite? Explain your answers.

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