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

In summary, the conversation is about trying to figure out a proof involving vectors and matrices. The goal is to prove that the inverse of a matrix A can be represented as I-au(w^T), where a is a scalar and u and w are vectors. To check if this is the inverse, one would need to verify if AB=BA=Identity, where B is equal to I-auwT.
  • #1
Chris Rorres
4
0
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)
 
Last edited:
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  • #2
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?
 

1. What is a Householder matrix?

A Householder matrix is a special type of matrix used in linear algebra. It is an orthogonal matrix that is symmetric and idempotent, meaning that when multiplied by itself, it results in the same matrix. It is often used in numerical linear algebra for solving systems of linear equations and for computing eigenvalues and eigenvectors.

2. How is a Householder matrix created?

A Householder matrix is created through a process called Householder transformation. This involves finding a reflection matrix that will reflect a vector onto the orthogonal complement of a given vector. The resulting matrix is then used to construct the Householder matrix.

3. What is the proof for the properties of a Householder matrix?

The proof for the properties of a Householder matrix involves using mathematical concepts such as matrix multiplication, orthogonality, and idempotency. The proof shows that the Householder matrix indeed possesses these properties and how they are derived from the Householder transformation process.

4. How is a Householder matrix used in practical applications?

A Householder matrix is used in various practical applications such as image and signal processing, data compression, and linear regression. It is also used in numerical linear algebra for solving systems of linear equations and computing eigenvalues and eigenvectors.

5. Are there any limitations or drawbacks of using a Householder matrix?

While Householder matrices have many useful properties, they can also be computationally expensive to compute, especially for large matrices. Additionally, they may not always be the most efficient choice for solving certain types of linear equations. However, these limitations can often be mitigated by using alternative matrix factorization methods.

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