Let A be an n x n symmetric matrix. Let {f₁,..., fn} be an orthonormal basis of eigenvectors of A satisfying Af; = Ajfj, for j = 1,..., n. n (a) Show that A = - Σλ;fff. j=1 compare the (i, j)-entry of n j=1 Aff with the (i, j)-entry of A = PDPT (b) Given polynomial p(x) = ax + ak-12k-1 + ax + ao, define the matrix p(A) = akAk +ak-1A-1 + a₁A+aoIn. Show that p(A₁),...,p(An) are the eigenvalues of p(A) and n p(A) = [p(\;)£;£³. j=1

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