Consider the symmetric matrix 0 -1 1 A = 0 −1 1 -1 0 a) Diagonalize the matrix A in the form A = SAST, with S an orthogonal matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) a. the rank of the matrix A. b. the determinant of the matrix A. C. the null space of the matrix A. c) Decompose the quadratic form Q(x) = x B x with B = A² and x = r = rank(B) squares of independent linear forms. = [x₁ X2 x3] as the sum of (Note: different solutions exist, one is sufficient! Either use the elimination method or the eigenvalue decomposition computed in a) where you don't need to explicitly compute B).

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