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I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here.

A theorem at the page linked states that if two matrices $A,B$ are non-singular and of dimension $r \times r$, then:

$$(A+B)^{-1} = B^{-1}(B^{-1}+A^{-1})A^{-1} = A^{-1}(A^{-1}+B^{-1})B^{-1} $$

Why is this true?

(a link to a proof will suffice).

Monolite
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    I would choose a different reference to learn decision theory. Just my idea. – Seyhmus Güngören Jan 27 '15 at 14:13
  • @SeyhmusGüngören I would be happy to hear your advice. – Monolite Jan 27 '15 at 14:18
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    http://www.cs.bme.hu/~gyorfi/testinghi.pdf for example but this is not complete at all. Only the things inside are relavent and modern. A complete book is by E. L. Lehman Testing Statistical Hypothesis. This book is however very heavy. Another book is a bit more engineering oriented but useful. Fundamentals of Statistical Signal Processing, Volume II: Detection Theory. This is easy to understand. – Seyhmus Güngören Jan 27 '15 at 14:35
  • https://math.stackexchange.com/q/1505628/321264 – StubbornAtom Jan 28 '22 at 13:37

1 Answers1

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$$(A+B)^{-1}=[B(B^{-1}A+I)]^{-1}=[B(B^{-1}+A^{-1})A]^{-1}$$

Now use that $(XY)^{-1}=X^{-1}Y^{-1}$ to get

$$[B(B^{-1}+A^{-1})A]^{-1}=A^{-1}(B^{-1}+A^{-1})^{-1}B^{-1}$$

Notice that there is a missing $-1$ in your formula.

To get the other equation start with

$$(A+B)^{-1}=[A(I+A^{-1}B)]^{-1}=[A(B^{-1}+A^{-1})B]^{-1}$$ and then

$$[A(B^{-1}+A^{-1})B]^{-1}=B^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}$$

Pp..
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