Homework SolutionEigenvalue of a Matrix: Proof Involving Nonsingular Matrices

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The discussion revolves around proving that if (I + A) is nonsingular, then (I - A)(I + A)⁻¹ = (I + A)⁻¹(I - A), establishing that (I - A)/(I + A) is defined. The initial proof attempts to show this by defining A and B, leading to the conclusion that A = B. However, a key point raised is the uncertainty regarding the invertibility of (I - A), which could invalidate the proof. The participant later clarifies that A is skew-symmetric, allowing the assumption that (I - A) is nonsingular. The discussion seeks to identify any remaining errors in the proof process.
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Proof involving nonsingular matrices.

Homework Statement



If (I + A) is nonsingular, prove that (I - A)(I + A)-1 = (I + A)-1(I - A), and hence (I - A)/(I + A) is defined for the matrix.

I've proved it like this:

Let (I - A)(I + A)-1 = A, and (I + A)-1(I - A) = B.
B-1 = (I - A)-1(I + A)
B-1A = I
Premultiplying by B, we get A = B.

Is this proof correct?
 
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chocolatefrog said:

Homework Statement



If (I + A) is nonsingular, prove that (I - A)(I + A)-1 = (I + A)-1(I - A), and hence (I - A)/(I + A) is defined for the matrix.

I've proved it like this:

Let (I - A)(I + A)-1 = A, and (I + A)-1(I - A) = B.
B-1 = (I - A)-1(I + A)
B-1A = I
Premultiplying by B, we get A = B.

Is this proof correct?

You don't know that I-A is invertible. So (I-A)-1 might not exist.
 


micromass said:
You don't know that I-A is invertible. So (I-A)-1 might not exist.

Oh, I forgot to mention that A is known to be skew-symmetric. So, (I - A)T = (I + A), which is nonsingular. And since a matrix is nonsingular iff its transpose is nonsingular, we could assume that (I - A)-1 exists.

I can't seem to think beyond this point. If there's still an error somewhere in the proof, could you please point to it?
 

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