MATRIX THEORY CHAPTER 5 FALL 2017 1.VECTOR NORMS (4④z+≤z+ (任，≥0 四在+=,)+(2, (⑤)）红，到=，或 Theorem 1(Cauchy-Schwarz inequality).If is an inner product on a vector space V over the field F CorR,then for all x,y∈V I红，P≤a,,以 Theorem2.f(,）is an inner product on a vector space V,then‖=√在，E)is a vector norm. Example 1.The IP norm on Cn is -( forl≤p≤In particular: z=+lz2+…+zn: l2=VP+2P+…+EnF Ilall=l. Theorem3.Suppose Virpace f finte dimension.Thenn.onV are equivalent in the sense that there erist constant C>such that 1/CIxl.≤lzl。≤Cl,VxeV 2.MATRIX NORMS Definition3.A function·l:Mn-→R is a Matrir norm if for any A,B∈Mn, (1)WAl≥0： (②)ⅢA=0 if and only if A=0:MATRIX THEORY - CHAPTER 5 FALL 2017 1. Vector Norms Definition 1. Let V be a vector space over a field F (C or R). A function k · k : V −→ R is a vector norm if for any x, y ∈ V , (1) kxk ≥ 0; (2) kxk = 0 if and only if x = 0; (3) kcxk = |c|kxk; (4) kx + yk ≤ kxk + kyk. Definition 2. Let V be a vector space over a field F (C or R). A function h·, ·i : V × V −→ F is an inner product if for any x, y, z ∈ V , (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hcx, yi = chx, yi for any c ∈ F; (4) hx + z, yi = hx, yi + hz, yi; (5) hx, yi = hy, xi. Theorem 1 (Cauchy-Schwarz inequality). If h·, ·i is an inner product on a vector space V over the field F (C or R), then for all x, y ∈ V |hx, yi|2 ≤ hx, xihy, yi. Theorem 2. If h·, ·i is an inner product on a vector space V , then kxk = p hx, xi is a vector norm. Example 1. The l p norm on C n is kxkp = Xn i=1 |xi | p !1 p for 1 ≤ p ≤ ∞ In particular: kxk1 = |x1| + |x2| + · · · + |xn|; kxk2 = p |x1| 2 + |x2| 2 + · · · + |xn| 2; kxk∞ = max 1≤i≤n |xi |. Theorem 3. Suppose V is a vector space of finite dimension. Then any two vector norms k · k?, k · k◦ on V are equivalent in the sense that there exist constant C > 0 such that 1/Ckxk? ≤ kxk◦ ≤ Ckxk?, ∀ x ∈ V. 2. Matrix Norms Definition 3. A function 9 · 9 : Mn −→ R is a Matrix norm if for any A, B ∈ Mn, (1) 9A9 ≥ 0; (2) 9A9 = 0 if and only if A = 0; (3) 9cA9 = |c| 9 A9; (4) 9A + B9 ≤ 9A 9 + 9 B9, (5) 9AB9 ≤ 9A 9 9B9. 1