Convexity, Schur convexity, and majorization theory - Signal Processing and Optimization for Transceiver Systems - page 727

Table 21.2. Majorization: summary

Some of the key points about majorization are summarized here; P denotes the size
of the vectors.

( y x )if k =0 y [ k ] ≥ k =0 x [ k ] , for 0 ≤ n ≤ P − 2 , and 1. y majorizes x

P− 1
k

y k = P− 1
k

x k (Definition 21.3).

=0 =0

0] T y P− 1 ] T

P ] T

2. [ 1 0 ... [ y 0 y 1 ... [ P

P

... for y k ≥ 0 ,

and k y k = 1 (Lemma 21.1).

if and only if P− 1
k

≥ P− 1
k

3. y x g ( y k ) g ( x k ) for all continuous convex

=0 =0

functions g ( x ) (Theorem 21.2).

4. y x if and only if x =( T 1 T 2 ... T P− 1 ) y for some sequence of T -transforms

T k (end of Sec. 21.5.2).

5. y x if and only if there exists a doubly stochastic matrix A such that x = Ay

(Theorem 21.9).

6. y x if and only if there exists an orthostochastic matrix A such that x = Ay

(Theorem 21.10).

7. A is doubly stochastic if and only if y majorizes Ay for every real vector y

(Theorem 21.8).

8. For Hermitian A , the vector of eigenvalues ( λ k ) majorizes the vector of diagonal
elements ( a kk ) (Theorem 21.5).
9. For Hermitian A , P− 1
k =0

1 / (1 + λ k ) ≥ P− 1
k =0

1 / (1 + a kk ) (Ex. 21.9).

10. For Hermitian matrices A 1 and A 2 , the sum of eigenvalues majorizes the eigen-
values of the sum, i.e., λ ( A 1 )+ λ ( A 2 ) λ ( A 1 + A 2 ) (Theorem 21.6).

Note: All functions, arguments, and constants are real-valued .