A∈Mn(R)A∈Mn(R) such that I+A\">I+AI+A is invertible. Suppos | StudyZin" /> A∈Mn(R)A∈Mn(R) such that I+A\">I+AI+A is invertible. Suppos | StudyZin" />
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Let A∈Mn(R)\">A∈Mn(R)A∈Mn(R) such that I+A\">I+AI+A is invertible. Suppose thatB=(I−A)(I+A)−1\">B=(I−A)(I+A)−1B=(I−A)(I+A)−1(a) Show that B=(I+A)−1(I−A)\">B=(I+A)−1(I−A)B=(I+A)−1(I−A) (b) Show that I+B\">I+BI+B is invertible and express A\">AA in terms of B\">BB.

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Basic Math

Let A∈Mn(R)\">A∈Mn(R) such that I+A\">I+A is invertible. Suppose that

B=(I−A)(I+A)−1\">B=(I−A)(I+A)−1

(a) Show that B=(I+A)−1(I−A)\">B=(I+A)−1(I−A) 

(b) Show that I+B\">I+B is invertible and express A\">A in terms of B\">B.

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Solution a Show thatBIA1IABIA1IABIA1IAWe    See Answer
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