Décomposition polaire

A∈ Mn(C)H∈H+
nA∗A=H2
A∈GLn(C) (U, H)∈Un×H+
n
A=UH
U
kA−Uk=d(A−Un)
M7→ ptr(M∗M)
M=A∗A M M
D=diag(λ1, . . . , λn)
P∈Un
M=P DP ∗
D= ∆2∆ = diag(√λ1, . . . , √λn)M=P∆2P∗= (P∆P∗)2
H=P∆P∗
H0
D
∆Q∈UnH0=Q∆Q∗
M=P DP ∗=H02=Q∆2Q∗=QDQ∗
Q∗P D =DQ∗P
Q∗P= (ai,j ) 1 ≤i, j ≤n aij λi=λjaij λi6=λjaij = 0
aij √λi=pλjaij Q∗P∆ = ∆Q∗P
P∆P∗=Q∆Q∗H=H0
A=UH U ∈UnH A∗A=HU∗UH =H2
H H
A U =AH−1
H A∗A H
A∗A U =AH−1U
U∗U=H−1A∗AH−1=H−1H2H−1=In
Mn(C)kAk=ptr(A∗A)
H=P∗∆P P ∆ = diag(µ1, . . . , µn)µi∈R+
kA−Uk2=kU(H−In)k2=kH−Ink2=tr(H−In)2=tr(∆ −I)2=
n
X
i=1
(µi−1)2
V∈Un
kA−Vk=kUH−Vk=kV−1UH−Ink=kV−1UP ∗∆P−Ink=kP(V−1UP ∗∆P−In)P∗k=kP V −1UP ∗∆−Ink
W=P V −1UP ∗
kA−Vk2=kW∆−Ink2=tr(∆W∗W∆−W∆−∆W∗+In)
=tr(∆2−W∆−∆W∗+In)
=
n
P
i=1
(µ2
i−(ωii +ωii)µi+ 1)
≥
n
P
i=1
(µ2
i−2µi+ 1) = kA−Uk2
P|ωii|2= 1 Re(ωii)≤ |ωii| ≤ 1
ωii = 1 i W =InV−1U=InV=U