Problem DI Let ME HIM"" be a symmetric matrix . As you have already learned in MAT Z Z A OT MIAT BT , an Eigenvector of A is a nonzero VECtorKE [ " such that Ax = AX for some MEL, the corresponding Eigenvalue . Here [ denotes the complex numbers ; i.e . all numbers of the form = = a + bi where a and bare real numbers and i = V' - I is the square root of - 1. Also , you will also need to know that E = Q - bi is the complex conjugate of & and if &, and Id = are any two complex numbers we have Zizz = 21 22 . In the following prob !` oblem , you may assume that all of the eigenvalues of' A are distinct. ( a) Prove that all of the eigenvalues of' A are real. (Hint : If' is a [ complex; eigenvalue of' A with eigenvector* , then it's complex conjugate A is also an eigenvalue of A with LIFEINVECtor*! ! ( b) Prove that if * and *2 are eigenvectors corresponding to distinct eigenvalues , then*' and `` are orthogonal