Let a i

j be the elements of a square matrix, not necessarily symmetrical, and let its inverse, b j

i satisfy a i

j b

j k = δ

i k = a j

k b

i j

. Show that the derivatives satisfy 1.7 Show that the spectral decomposition of the the symmetric matrix A A = QΛQT

, QQT = I, Λ = diag {λ1,…, λp}

is unique up to permutations of the columns of Q and the elements of Λ if the eigenvalues of A are distinct.