Definition Definition A vector norm is a function from Cn to R with three properties: Nonnegative:x≥0 for all x∈Cn, x=0 if and only if x =0. Homogeneous:lax]ax]I for all a C,x C. 奇电有这头 Matrix Theory Vector Norms -5/39
Definition Definition A vector norm ∣∣ ⋅ ∣∣ is a function from C n to R with three properties: Nonnegative: ∣∣x∣∣ ≥ 0 for all x ∈ C n , ∣∣x∣∣ = 0 if and only if x = 0. Homogeneous: ∣∣αx∣∣ = ∣α∣∣∣x∣∣ for all α ∈ C, x ∈ C n . Triangle inequality: ∣∣x + y∣∣ ≤ ∣∣x∣∣ + ∣∣y∣∣ for all x, y ∈ C n . Matrix Theory Vector Norms - 5/39
Definition Definition A vector norm is a function from Ch to R with three properties: Nonnegative:x≥0 for all x∈Cn, llxll =0 if and only if x=0. Homogeneous:llaxll alxll for all a C,xC. Triangle inequality:x +yll sx+llyll for all x,y e C. 奇电有头子 Matrix Theory Vector Norms -5/39
Definition Definition A vector norm ∣∣ ⋅ ∣∣ is a function from C n to R with three properties: Nonnegative: ∣∣x∣∣ ≥ 0 for all x ∈ C n , ∣∣x∣∣ = 0 if and only if x = 0. Homogeneous: ∣∣αx∣∣ = ∣α∣∣∣x∣∣ for all α ∈ C, x ∈ C n . Triangle inequality: ∣∣x + y∣∣ ≤ ∣∣x∣∣ + ∣∣y∣∣ for all x, y ∈ C n . Matrix Theory Vector Norms - 5/39
Definition Vector p-Norms The unit "circle"considered under different p-norms. 务老环这女子 Matrix Theory Vector Norms -6/39
Definition Vector p-Norms The unit “circle” considered under different p-norms. Matrix Theory Vector Norms - 6/39
Definition Vector p-Norms The unit "circle"considered under different p-norms. The vector p-norms are useful for computational purposes,as well as analysis. 务老环这女子 Matrix Theory Vector Norms -6/39
Definition Vector p-Norms The unit “circle” considered under different p-norms. The vector p-norms are useful for computational purposes, as well as analysis. Matrix Theory Vector Norms - 6/39
Definition Vector p-Norms Let x e Cn with elements x =(x1...,xn)T The p-norm The unit"circle"considered under lxp=(∑1P)Vp,p≥1， different p-norms. The vector p-norms are useful for is a vector norm. computational purposes,as well as analysis. 务老环这女子 Matrix Theory Vector Norms -6/39
Definition Vector p-Norms The unit “circle” considered under different p-norms. The vector p-norms are useful for computational purposes, as well as analysis. Let x ∈ C n with elements x = (x1, . . . , xn) T . The p-norm ∣∣x∣∣p = (∑ n j=1 ∣xj ∣ p ) 1/p , p ≥ 1, is a vector norm. Matrix Theory Vector Norms - 6/39