Читать книгу Artificial Intelligence and Quantum Computing for Advanced Wireless Networks - Savo G. Glisic - Страница 91

(Properties of L_sym and L_rw ) The normalized Laplacians satisfy the following properties:

1 For every f ∈ ℝn we have

2 λ is an eigenvalue of Lrw with eigenvector u if and only if λ is an eigenvalue of Lsym with eigenvector w = D1/2 u.

3 λ is an eigenvalue of Lrw with eigenvector u if and only if λ and u solve the generalized eigen problem Lu = λDu.

4 0 is an eigenvalue of Lrw with the constant one vector I as eigenvector. 0 is an eigenvalue of Lsym with eigenvector D1/2I.

5 Lsym and Lrw are positive semidefinite and have n non‐negative real‐valued eigenvalues 0 = λ1≤,….≤λn.

Proof. Part (1) can be proved similarly to Part (1) of Proposition 5.1.

Part (2) can be seen immediately by multiplying the eigenvalue equation L_sym w = λw with D^−1/2 from the left and substituting u = D^−1/2 w.

Part (3) follows directly by multiplying the eigenvalue equation L_rw u = λu with D from the left.

Part (4): The first statement is obvious as L_rwI = 0, the second statement follows from (2).

Part (5): The statement about L_sym follows from (1), and then the statement about L_rw follows from (2).

Part (5): The statement about L_sym follows from (1), and then the statement about L_rw follows from (2).