Problem 1. (Classifying non-diagonalizable 2 x 2 matrices.) Let A € R²x2 be a 2 × 2 matrix. (a) Suppose that A has eigenvalue 0 but is not diagonalizable. Prove that im(A) = Eo, and conclude from this...

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Extracted text: Problem 1. (Classifying non-diagonalizable 2 x 2 matrices.) Let A € R²x2 be a 2 × 2 matrix. (a) Suppose that A has eigenvalue 0 but is not diagonalizable. Prove that im(A) = Eo, and conclude from this that A2 = 0. (b) Let A ER and suppose that A has eigenvalue A but is not diagonalizable. Prove that we have (A – Al2)? = 0, and deduce from this that Au – AT e E, for every i e R². [Hint: apply part (a) to the matrix A – AI2.] (c) Prove that if A has eigenvalue A but is not diagonalizable, then A is similar to [Hint: consider the basis B = (A® – Aū, T) where ī ¢ E,.] (d) Prove that if A does not have any real eigenvalues, then A is similar to a matrix of the form AQ where Q is an orthogonal matrix and A > 0. 'We work over R throughout this problem. So“eigenvalue" means real eigenvalue, "diagonalizable" means diag- onalizable over IR, and "similar" means similar over R. 2Recall that for each A E R, Ex = {v € R? : Au = Au}.