# 程序代写代做代考 handoutC.dvi

handoutC.dvi

ECS130 Scientific Computation Handout C January 20, 2017

1. Norms are an indispensable tool to provide vectors and matrices with measures of size, length
and distance.

2. A vector norm on Cn is a mapping that maps each x ∈ Cn to a real number ‖x‖, satisfying

(a) ‖x‖ > 0 for x 6= 0, and ‖0‖ = 0 (positive definite property)
(b) ‖αx‖ = |α| ‖x‖ for α ∈ C (absolute homogeneity)
(c) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ (triangle inequality)

3. Commonly used vector norms:

‖x‖1 =
n

i=1

|xi|, “Manhattan” or “taxi cab” norm

‖x‖2 =
(

n

i=1

|xi|2
)

1/2

=

xHx, Euclidean length

‖x‖∞ = max
1≤i≤n

|xi|.

4. The geometry of the closed unit “ball”: {x ∈ C2 : ‖x‖p ≤ 1} for p = 1, 2,∞.

5. Norm equivalence: Let ‖·‖α and ‖·‖β be any two vector norms. There are constants c1, c2 > 0
such that

c1‖ · ‖α ≤ ‖ · ‖β ≤ c2‖ · ‖α
For examples, it can be easily shown that

‖x‖∞ ≤ ‖x‖2 ≤

n‖x‖∞

‖x‖2 ≤ ‖x‖1 ≤

n‖x‖2

‖x‖∞ ≤ ‖x‖1 ≤ n‖x‖∞

6. Cauchy-Schwarz inequality:
|xHy| ≤ ‖x‖2‖y‖2

with equality if and only if x and y are linearly dependent.

7. A matrix norm on Cm×n is a mapping that maps each A ∈ Cm×n to a real number ‖A‖,
satisfying

(a) ‖A‖ > 0 for A 6= 0, and ‖0‖ = 0 (positive definite property)
(b) ‖αA‖ = |α| ‖A‖ for α ∈ C (absolute homogeneity)
(c) ‖A+B‖ ≤ ‖A‖+ ‖B‖ (triangle inequality)

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8. Example: for A = (aij) ∈ Cm×n, the Frobenius norm ‖A‖F is defined by

‖A‖F
def
=

m

i=1

n

j=1

|aij |2

1/2

=

tr(AHA).

9. The induced matrix norm ‖ · ‖:
A vector norm ‖ · ‖ induces a matrix norm, denoted by the same notation:

‖A‖ def= max
x 6=0

‖Ax‖
‖x‖

= max
‖x‖=1

‖Ax‖

(Exercise. verify that ‖A‖ is indeed a norm on Cm×n

10. Useful property: ‖Ax‖ ≤ ‖A‖ ‖x‖. Therefore, ‖A‖ is the maximal factor by which A can
“strech” a vector.

11. The vector p-norms induce the matrix p-norms, in particular, for p = 1, 2,∞, we have

‖A‖1 = max
x 6=0

‖Ax‖1
‖x‖1

= max
1≤j≤n

{

m

i=1

|aij |
}

= max absolute column sum,

‖A‖2 = max
x 6=0

‖Ax‖2
‖x‖2

= the largest singular value of A,

‖A‖∞ = max
x 6=0

‖Ax‖∞
‖x‖∞

= max
1≤i≤m

n

j=1

|aij |

= max absolute row sum.

12. An application: sensitivity analysis of linear system of equations Ax = b.

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