Question

In: Math

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

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

B=(IA)(I+A)1">B=(I−A)(I+A)−1

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

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

Solutions

Expert Solution

Solution

(a) Show that B=(I+A)1(IA)">B=(I+A)−1(I−A) We have(I+A)(IA)=IA+AA2(IA)(I+A)=I+AAA2(I+A)(IA)=(IA)(I+A)(IA)=(I+A)1(IA)">(I+A)(I−A)=I−A+A−A2(I−A)(I+A)=I+A−A−A2(I+A)(I−A)=(I−A)(I+A)(I−A)=(I+A)−1(I−A)Muhiply from the right side by (I+A)1">(I+A)−1 

then we get   (IA)(I+A)1=(I+A)1(IA)">(IA)(I+A)1=(I+A)1(IA)">(I−A)(I+A)−1=(I+A)−1(I−A) 

Therefore, B=(I+A)1(IA)">B=(I+A)−1(I−A) 

(b) Show that I+B">I+B is invervible and express A">A in terms of B">B We have(I+A)(I+B)=(I+A)+(I+A)B=I+A+IA=2I">(I+A)(I+B)=(I+A)+(I+A)B=I+A+I−A=2ITherefore, (I+B)">(I+B) is invertible and its inverse is 12(I+A)$Then">12(I+A)$Then">12(I+A)$Then(IB)(I+B)1=12(IB)(I+A)=12[I+AB(I+A)]=12(I+AI+A)(IB)(I+B)1=A">(I−B)(I+B)−1=12(I−B)(I+A)=12[I+A−B(I+A)]=12(I+A−I+A)(I−B)(I+B)−1=A$


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