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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