Re: [線代] 正定矩陣的問題
※ 引述《mrporing (波利先生)》之銘言：
: 有個證明題是這樣的:
: A為一個n*n的正定實數方陣，B是一個m*n的實數矩陣且為滿秩(full rank)，則證明
: BAB^(T)也會是正定。
(i) Obviously, if m > n, it is not true
(ii) if m <= n, it is yes
Proof of (ii)
if not, then exists a nonzero row vector x such that
x.B.A.B^T.x^T = 0
denote y = x.B
then y.A.y^T = 0
but A is positive definite
hence, y = 0
that is, x.B = 0 has a nontrivial solution.
Contradiction.
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