※ 引述《magnolia0922 (MagnoliaLover)》之銘言：
: A is diagonal dominant nxn matrix.
: Prove that A' is also diagonal dominant,where A' is the resulting matrix after
: first step Gaussian elimination without pivoting.
: 求解
: 下面是我自己推 但是寫到一半不知道怎麼寫下去了
: http://i.imgur.com/5xdEUew.jpg
If a_21 = 0, the conclusion is obvious.
WLOG a_11 = 1 and a_21 = 1 (can divide by a_11 and a_21 for the first and
the second rows, respectively)
So now we have
1 >= |a_12| + ... + |a_1n|
and
|a_22| >= 1 + |a_23| + ... + |a_2n|
Then |a_22 - a_12|
>= |a_22| - |a_12|
>= 1 + |a_23| + ... + |a_2n| - |a_12|
>= 1 + |a_23 - a_13| - |a_13| + |a_24 - a_14| - |a_14| + ...
+ |a_2n - a_1n| - |a_1n| - |a_12|
>= |a_23 - a_13| + ... + |a_2n - a_1n|
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 97.99.68.240
※ 文章網址: https://www.ptt.cc/bbs/Math/M.1488656392.A.098.html
※ 編輯: cuttlefish (97.99.68.240), 03/05/2017 03:41:15