How can the inverse of a Householder matrix be verified using the given proof?

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The discussion focuses on verifying the inverse of a Householder matrix defined as A = I + u(w^T), where u and w are vectors in ℝⁿ, and I is the identity matrix. The proof requires showing that A^-1 = I - a u(w^T), with a = 1/(1 + (w^T)u), under the condition that (w^T)u ≠ -1. To confirm that B = I - a u(w^T) is indeed the inverse of A, one must demonstrate that the product AB equals the identity matrix.

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Chris Rorres
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I'm working on trying to figure this proof out but its proving to be quite difficult does anyone have any insight?

Let u and w be vectors in (all real numbers)^n, and let I denote the (n × n) identity matrix. Let A= I + u(w^T), and assume that (w^T)u doesn’t equal -1 (notice that (w^T)u produces a scalar). Prove that
A^-1= I–au(w^T), where a = 1/(1+(w^T)u)
 
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The definition of the inverse of A is B so that AB=BA = Identity

So if B=I-auwT, what should you do to check that B is the inverse of A?
 

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