矩阵论 第五章 Hermite矩阵和正定矩阵.pdf

Hermite 5.1 HermiteHermite 5.1.1 Hermite • 2 3 − 2i , −1 7 3 + 2i −5 7 4 • Hermite (1) A Hermite k A k Hermite (2) A Hermite A −1 Hermite (3) A B Hermite k p kA + pB Hermite (4) A B Hermite AB Hermite AB = BA H (5) A Hermite S S AS Hermite • 5.1.1 A = (ajk ) ∈ Cn ×n A Hermite n H x ∈ C x A x • 5.1.2 A n Hermite (1) A (2) A (2) λµ A x y A x = λx (1) A y = µy (2) (??) H H y A x = λy x (??),µ = µHA = A H H H H H y A x = y A x = µy x H (λ − µ)y x = 0 λ = µ x y • 5.1.3 A ∈ Cn ×n A Hermite U H U AU = Λ = diag(λ , λ , · · · , λ ) 1 2 n λ , λ , · · · , λ 1 2 n • 5.1.4 A ∈ Rn ×n A Q T Q AQ = diag(λ , λ , · · · , λ ) 1 2 n λ , λ , · · · , λ 1 2 n 5.1.2 • 1.6.8 A,B ∈ Cn ×n nP B = PHAP A B • 5.1.5 A n Hermite A I s 0 0    D0 = 0 −Ir−s 0 0 0 0n−r r =rank (A )s A () 5.1.3 n U