Esame Calcolo numerico I

Norme di un vettore

Le norme di un vettore possono essere espresse in diversi modi:

1. Norma Euclidea: ||x||₂ = sqrt(sum(|x_i|²))
2. Norma uno: ||x||₁ = sum(|x_i|)
3. Norma infinito: ||x||_∞ = max(|x_i|)

Tutte e tre fanno parte di una famiglia generale di norme, definite come:

||x||_p = (sum(|x_i|^p))^1/p, dove p ≥ 1.

Da notare che quando p tende all'infinito, si ha:

||x||_∞ = lim_p→∞ (sum(|x_i|^p))^1/p = max(|x_i|)

Quindi, per ogni componente si ha che |x_i| ≤ ||x||_∞.

Inoltre, si può dimostrare che:

1||x|| ≤ ||x|| ≤ ||x||_n^∞

E visto che lim_n→∞ ||x||_n = ||x||_∞, si ha:

||x||_∞ = max_i=1≤i≤n |x_i|

Per determinare quanto più grande potrebbe essere ||y|| in confronto a ||x||, si può scrivere:

||y|| / ||x|| = ||x||_∞

Il fattore di espansione della norma è quindi:

||M|| = sup(||x||_∞ / ||x||), dove x ≠ 0

Il massimo fattore di espansione della norma è dato dalla formula:

||M|| = max y ||Mx|| / ||x|| = max y ||M|| ||x|| / ||x|| = max y ||M||

dove y è un vettore unitario.

Per due matrici A e B, si ha:

||A + B|| = max (A + B)x = max (A + B)x / ||x|| = max (A + B)x / ||x|| = max (A + B)

Un'altra proprietà importante è:

||Mx|| / ||x|| ≤ ||M||

Abbiamo visto che:

||x - x̂|| = ||A|| ||r||

Per la precisione, x̂ è chiamato errore assoluto. Spesso è più utile usare l'errore relativo:

||x - x̂|| / ||x||

che fornisce l'errore come una percentuale della norma della risposta.

esatta.8Visto che b = Ax ||A||1||b|| ||Ax|| ≤ ||A|| × ||x|| ≤= e ||x|| ||b||||A|| ||A||1 −1||x − × ≤ ||x − × ≤ ||A || ||r|| ×=⇒ x̂|| x̂||||x|| ||b|| ||b||||r||||x − x̂|| −1≤ ||A|| ||A ||=⇒ ||x|| ||b||||r|| ||b− b̂||− −Possiamo scrivere r = b Ax̂ = b b̂ e quindi = è l’errore relativo nel termine noto.||b|| ||b||−1||A|| ||A || è il gattore di espansione dell’errore relativo, si chiama numero di condizionamento dellamatrice.8 Formule per Norme di Matrici8.1 Teorema per la norma infinitom×n∈Consideriamo A . AlloraR nX |A |||A|| = max∞ ij1≤i≤m j=1cioè la massima somma delle righe in valore assolutoDIM:||Ax|| ||Ax||= max .∞ ∞||x||∞=1n∈ ||x|| |x | ≤Sia x , = 1 =⇒ 1∀jR ∞ j nX||Ax|| A x(Ax) = max= max∞ ij ji 1≤i≤m1≤i≤m j=1 nn XX |A ||A | |x | ≤≤ maxmax ij j

ij1≤i≤m1≤i≤m j=1 j=1||A|| ||Ax|| ≥ ||Aa|| ∀a ∈ ||a||= max = 1.R∞ ∞ ∞ ∞||x||∞=1

Consideriamo un a speciale; sia k l’indice di una rigan nX X|A | |A |= maxkj ij1≤i≤mj=1 j=1

Definiamo:

+1 se A > 0kj

a := sgn(A ) = 0 se A = 0j kj kj

−1 se A < 0 kj

||a||Notiamo che = 1. Quindi∞ ||A|| ≥ ||Aa|| ≥= max (Aa) (Aa)∞ ∞ i k1≤i≤mn nX X= A a = A sgn(A )kj j kj kjj=1 j=1n nX X|A | |A |= = maxkj ij1≤i≤mj=1 j=1

Abbiamo dimostrato che n nX X|A | ≤ ||A|| ≤ |Amax max∞ij ij1≤i≤m 1≤i≤mj=1 j=1e quindi nX||A|| |A |= max∞ ij1≤i≤m j=1

8.2 Teorema per la norma unom×n∈Consideriamo A . AlloraR mXt||A|| ||A || |A |= = max∞1 ij1≤j≤n i=1cioè, la massima somma delle colonne in valore assoluto.

8.3 Teorema: Dualità n∈Abbiamo le seguenti formule duali per le norme di un vettore x Rt||x|| = max y x1 ||y|| =1∞ t||x|| = max y

x² ||y|| = 12 t||x|| = max y x∞ ||y|| = 11

DIM: # NORMA 1

n∈ ||y||

Sia y tc = 1R ∞ n n nX X Xt ≤ |y | × |x | ≤ × |x | ||x||y x = y x 1 =i i i i i 1i=1 i=1 i=1t

Quindi max y x.||y|| = 1∞ t t n≥ ∈ ||a||

Dall’altra parte max y x a x per qualsiasi candidato a = 1.R ∞||y|| = 1∞n∈ ||a||

Sia a dove a := sgn(x ) = 1.R ∞i i

Quindi n nX Xt t≥ |x | ||x||max y x a x = sgn(x )x = =i i i 1||y|| = 1∞ i=1 i=1

e abbiamo dimostrato che t||x|| ≤ ≤ ||x||max y x1 1||y|| = 1∞

10DIM: # PER NORMA 1 DI UNA MATRICE

t||A|| ||A ||

Useremo le formule di dualità per dimostrare che = ∞

1 t||A|| ||Ax||= max = max max y (Ax)1 1||x|| ||x|| ||y||=1 = 1 = 1∞

1 1 t t= max max y (Ax)||x|| ||y||=1 = 1∞

1 t t= max max x (A y)||x|| ||y||=1 = 1∞

1 t t= max max x (A y)||y|| ||x||=1 = 1∞

1t t||A ||A ||= max y|| =∞ ∞||y|| = 1∞

8.4 Teorema: Norma 2 du una matricem×n∈

Sia A . AlloraR p t||A|| ρ(A A)=2n×n∈dove,

Per una matrice M, il raggio spettrale |λ|ρ(M) è dato da max_1≤i≤n|λ_i|.

DIM: n×n ∈

Consideriamo prima il caso più semplice, A diagonale

R||A|| = max₂2||x|| = 1/2

pp t t t(Ax) (Ax) = max_x(A A)x = max_||x||1/2||x||² = 1/2

( )

nX 2 2= max_{A xi}i||x|| = 1/2 i=1

≥

Interpretazione: sia λ := x₀i i v nuXu 2

max λ A media pesata

it iinP λ = 1

ii=1 i=1

Quindi v nuXu 2 |A |max λ A = max_iiit iinP 1≤i≤n

λ = 1

ii=1 i=1

Allora, per una matrice diagonale ||A|| = |A| = max₂ii 1≤i≤n

DIM: # CASO GENERALE

Def -1n×n t t t∈P è ortogonale se P = P^T, cioè P P^T = P^T P = I. Sono matrici che rappresentano rotazioni

R n∈ o riflessioni; Ha la proprietà speciale che per x ∈ R²

t t t t||P ||x|| = (P x) (P x) = x (P P^T)x = x x = 1.

p t||A|| ||Ax|| = max = max (Ax) (Ax)

2 2||x|| ||x||=1 = 1/2

p t tx (A A)x = max||x|| = 1/2

t tMa M := A A è simmetrica; M = P DP dove P è ortogonale e D è la matrice diagonale

degli autovalori.

$$||A|| = \max_{x} \left(\frac{P^T D P}{||x||}\right) = \max_{(P x)} \left(\frac{D(P x)^2}{||x||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2} \max_{y} \left(\frac{Dy^T P^T P y}{||y||}\right) = \frac{1}{2}

saràR t t t t tx Ax = x (P DP )x = (P x) D(P x) = y Dy

[ ] λ 0 0 ... 0 y1 10 λ 0 ... 0 y n2 2

[ ] Xt 2

[ ] * * ∀ ∈ ≠ ⇔ ≤ ≥y ... y ...x Ax = = λ y > 0 y = 0 λ > 0 1 i n1 n i i

[ ] [ ] i[ ] [ ] * *... i=1

[ ] [ ] 0 ... 0 0 λ yn novvero A è definita positiva sse A simmetrica e i suoi autovalori sono tutti positivi.

9.1 Teorema: Choleskin×n n×n∈ ∈La matrice A è definita positiva sse esiste una matrice L triangolare inferiore,R Rnon-singolare tc t×A = L LLa matrice L si chiama fattore di Choleski di A.DIM con esempio... 129.1.1 Lemma n×n n+1×n+1∈ ∈La sottomatrice principale A di una matrice A definita positiva è anch’essaR R1definita positiva. A c1 −1n+1×n+1 t∈ −Se A = è definita positiva, allora a c (A ) > 0.R n+1,n+1 1tc an+1,n+1 ...10 Interpolazione