The Spectral Decomposition:
(i) Let A be a symmetric matrix with eigenvalues λ1,... ,λn and corresponding orthonormal eigenvectors μ1, ...,μ1. Let Pk = uk uTk be the orthogonal projection matrix onto the eigenline spanned by uk, as defined in Exercise 5.5.8. Prove that the spectral factorization (8.32) can be rewritten as
A = λ1 P1 + λ2 P2 + ... + λn Pn
= λ1 μ1 uT1 + λ2 u2 uT2 + ... + λn un uTn,
(8.34)
expressing A as a linear combination of projection matrices.
(ii) Write out the spectral decomposition (8.34) for the matrices in Exercise 8.4.14.
Matrices in Exercise 8.4.14
(a)

(b)

(c)

(d)

8 (3 1) 4 011 121 102 2