Proving Reversibility of A with u & v of Size n*1

In summary, the conversation discusses a proof involving u and v of size n*1 and I of size n*n. The proof shows that if u^T*v is not equal to -1, then A = I + u*v^Transpose is reversible and its inverse is A^-1 = I - (1 / (1+u^T*V))*uv^T. The conversation also mentions the importance of multiplying (I+uv^T)(I- \frac{1}{1+ u^Tv}uv^T) and (I- \frac{1}{1+ u^Tv}uv^T)(I+ uv^T) to show that they both result in the identity matrix.
  • #1
John Smith
6
0
I need help with this proof.
We have u and v of size n*1. It is giving that I of size n*n.
A = I + u*v^Transpose

Proof that if u^T*v is not = -1
then A is reverseble and that A is
A^-1 = I - (1 / (1+u^T*V))*uv^T
 
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  • #2
The obvious thing to do is to look at
[tex](I+uv^T)(I- \frac{1}{1+ u^Tv}uv^T)[/tex]
and
[tex](I- \frac{1}{1+ u^Tv}uv^T)(I+ uv^T)[/tex]

What do you get when you multiply those out?

(Are you certain that one of those [itex]uv^T[/itex] is not supposed to be [itex]vu^T[/itex]?)
 
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  • #3
I did multiply the equation out and find out that ((v^T)* u)^T = u^T * v
But I was wondering if this was enough to show out the proof.
 
  • #4
No, that is not the point at all!
The products of both
[tex](I+uv^T)(I- \frac{1}{1+ u^Tv}uv^T)[/tex]
and
[tex](I- \frac{1}{1+ u^Tv}uv^T)(I+ uv^T)[/tex]
should be the identity matrix!
 
  • #5
Yes thank you I think that I got it now.
 

What is the meaning of "Proving Reversibility of A with u & v of Size n*1"?

"Proving Reversibility of A with u & v of Size n*1" refers to the process of determining whether a matrix A can be reversed or inverted using vectors u and v of size n*1. This involves analyzing the properties of the matrix and the vectors to determine if A can be reversed and if so, what values of u and v are needed.

Why is proving the reversibility of A with u & v important?

Proving the reversibility of A with u & v is important because it allows for the use of matrix inversion, which is a fundamental operation in linear algebra. This process is used in many applications, such as solving systems of linear equations, finding eigenvalues and eigenvectors, and performing transformations in 3D graphics.

What factors affect the reversibility of A with u & v?

The main factors that affect the reversibility of A with u & v are the dimensions of the matrix and vectors, the properties of the matrix (such as being square and non-singular), and the operations performed on the matrix and vectors.

What techniques are used to prove the reversibility of A with u & v?

There are various techniques that can be used to prove the reversibility of A with u & v, such as Gaussian elimination, LU decomposition, and the use of determinants and inverse matrices. These techniques involve manipulating the matrix and vectors to determine if a solution exists.

What are the practical applications of proving the reversibility of A with u & v?

The practical applications of proving the reversibility of A with u & v are numerous. As mentioned before, it is used in solving systems of linear equations, finding eigenvalues and eigenvectors, and performing transformations in 3D graphics. It is also used in data compression, signal processing, and cryptography, among others.

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