Problem 1.2 Let a be some real number. Consider the following 4 X 4 matrix: A: colLIL colLIL QI'OO ILQDD (a) Find the spectral decomposition of A, that is write A = UAUT Where U is orthogonal and A is diagonal. (b) Use the eigenvalues of A to determine its rank. Him: The answer depends on whether |oz| = 1 or not. ((3) Assume that o: = 1 and nd an orthonormal basis for Im(A). (d) Assume that o: = 1 and nd an orthonormal basis for ker(A)? What is the dimension of ker(A)? (e) For what values of a is the matrix PSD