→ wohtp :先log兩邊會不會比較容易？ 08/15 13:15
→ wohtp :用 log det = tr log 08/15 13:15
※ 引述《Ohwil (LDPC)》之銘言：
: -1 -1 -1 -1
: (A+B) + (A-B) = 2(A - B A B )
: 想問一下這個式子是怎麼推導的
: 跟Woodbury matrix identity 有關?
: 出處 tao http://goo.gl/H8JMR
In the following, ' means ^(-1), x meansε.
det(A+B)det(A-B)
=det(B)det(1+B'A)det(A)det(1-A'B)
=det(B)det(A)det(1+B'A-A'B-1)
=det(B)det(A)det(B'A-A'B)
=det(B)det(AB'A-B)
i.e. det(A+B)det(A-B)=det(B)det(AB'A-B)
det(A+xH+B)det(A+xH-B)=det(B)det((A+xH)B'(A+xH)-B)
Extracting the linear component of x
tr((A+B)'H+(A-B)'H)
=tr((AB'A-B)'(HB'A+AB'H))
=tr((B'A(AB'A-B)'+(AB'A-B)'AB')H)
=tr(2(A-BA'B)H)
Since H is arbitrary,
we have (A+B)'+(A-B)'=2(A-BA'B)
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