高等微積分....急

Question

Construct an example of a matrix A such that for any of the operator

norms, ‖A‖^(-1)<‖A^(-1) ‖

A 2*2 diagonal example will suffice

不好意思~

麻煩可以寫的詳細一點嗎

真的是很感激不敬

太謝謝你啦...

Answer 1

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所有的 operator norm 都是等價的,

(http://en.wikipedia.org/wiki/Operator_norm)

我選 ‖A‖= max{|Av| : |v|=1} 這個.

取二階對角方陣 A = diag(3,4)

=> ‖A‖= max{|3v_1 + 4v_2| : (v_1)^2 + (v_2)^2 = 1}

由柯西不等式可知 |3v_1 + 4v_2| 的最大值是 5, 故‖A‖^(-1) = 1/5.

A^(-1) = diag(1/3,1/4)

=>‖A^(-1)‖= max{|(v_1)/3 + (v_2)/4| : (v_1)^2 + (v_2)^2 = 1}

由柯西不等式可知 |(v_1)/3 + (v_2)/4| 的最大值是 5/12.

故‖A‖^(-1) = 1/5 < 5/12 =‖A^(-1)‖.