推 HP0:謝謝解答！ 09/21 19:15
※ 引述《HP0 (cksh)》之銘言：
: Suppose that A and B are both symmetric positive definite matrices.
: Then all eigenvalues of AB are positive.
: 請問這個要怎麼證明？？
取x不等於0
H H
x A x > 0 x B x > 0
設Ax=入a x bx=入b x
H H H
x A x = x 入a x = x x 入a = |x| 入a>0
因為x不等於0 所以 入a>0
同理 入b>0
則ABx =A 入b x = 入b A x= 入b 入a x
得知 AB 的入 >0
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