Numerical Methods Contents - SAM

Remark 1.2.1 (Row-wise & column-wise view of matrix product).
irrelevant matrix entries
A ∈ K m,n , B ∈ K n,k :
)
AB =
(A(B) :,1 ... A(B) :,k
↓
matrix assembled from columns
⎛
, AB = ⎝ (A) ⎞
1,:B
. ⎠ .
(A) m,: B
↓
matrix assembled from rows
(1.2.2)
m
n
0
n
=
k
m
△
k
Remark 1.2.2 (Understanding the structure of product matrices). A “mental image” of matrix multiplication
is useful for telling special properties of product matrices
Ôº¿¿ ½º¾
⎛
⎜
⎝
⎞ ⎛
⎟ ⎜
⎠ ⎝
⎞ ⎛
=
⎟ ⎜
⎠ ⎝
⎞
,
⎟
⎠
Ôº¿ ½º¾
⎛
⎞ ⎛
⎞
⎛
⎞
m
n
0
n
= 0
k
m
⎜
⎝
⎟ ⎜
⎠ ⎝
=
⎟ ⎜
⎠ ⎝
.
⎟
⎠
m
0
k
n
=
0
m
Code 1.2.3: visualizing structure of matrices
1 n = 100; A = [ diag ( 1 : n−1) , ( 1 : n−1) ’ ; ( 1 : n ) ] ; B = A( n : − 1 :1 ,:) ;
2 C = A∗A ; D = A∗B ;
3 figure ; spy (A, ’ r ’ ) ; axis o f f ; p r i n t −depsc2 ’ . . / PICTURES/ Aspy . eps ’ ;
4 figure ; spy (B, ’ r ’ ) ; axis o f f ; p r i n t −depsc2 ’ . . / PICTURES/ Bspy . eps ’ ;
5 figure ; spy (C, ’ r ’ ) ; axis o f f ; p r i n t −depsc2 ’ . . / PICTURES/ Cspy . eps ’ ;
6 figure ; spy (D, ’ r ’ ) ; axis o f f ; p r i n t −depsc2 ’ . . / PICTURES/ Dspy . eps ’ ;
n
k
k
Ôº¿ ½º¾
△
Ôº¿ ½º¾