Eigenvalue Factorization and Matrix Substitution

In summary, the conversation discusses attempting to understand an equation involving eigenvalue factorization and the use of matrix U. The speaker mentions trying to change the equation to (A^T)A and working backwards, but struggles with the use of inverses. They also ask for clarification on the exponent of -2 in the context of a 4x4 matrix and the purpose of matrix U. The responder explains that U is the invertible matrix used in diagonalization and provides a resource for further understanding.
  • #1
the_dialogue
79
0
In my literature reviews I found a few things that I can't quite understand.

Homework Statement



I have the following equation:
http://img717.yfrog.com/img717/6416/31771570.jpg

I'm told that by using the eigenvalue factorization:
http://img89.yfrog.com/img89/760/83769756.jpg

, I can change the first equation to:
http://img28.imageshack.us/img28/5023/84802099.jpg

2. The attempt at a solution

I tried changing Equation 2 to just be (A^T)A and then subbing into the first equation, but I can't quite do anything with those inverses.

Also, what does the exponent of '-2' mean in the context of a 4x4 matrix? Lastly, what is matrix U?

Thank you!
 
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  • #2
the_dialogue said:
I tried changing Equation 2 to just be (A^T)A and then subbing into the first equation, but I can't quite do anything with those inverses.

I think it's probably easiest to start from [itex]\textbf{p}^{T}(\mathbf{\Lambda}+\lambda\textbf{I})^{-2}\textbf{q}=0[/itex] and work your way backwards instead.

Also, what does the exponent of '-2' mean in the context of a 4x4 matrix? Lastly, what is matrix U?

[tex]\textbf{C}^{-2}\equiv\textbf{C}^{-1}\textbf{C}^{-1}[/tex]

You simply square the inverse of the matrix.
 
  • #3
I'll give it a try gabbagabbahey. Thanks.

Any idea what the matrix "U" is?
 
  • #4
the_dialogue said:
Any idea what the matrix "U" is?

It's the invertible matrix which relates the matrix [itex]\textbf{A}^{T}\textbf{A}\mathbf{\Sigma}[/itex] to the diagonal matrix [itex]\mathbf{\Lambda}[/itex] via a similarity transform. Its columns will be the eigenvectors of [itex]\textbf{A}^{T}\textbf{A}\mathbf{\Sigma}[/itex].

See http://en.wikipedia.org/wiki/Diagonalizable_matrix for a refresher on matrix diagonalization.
 
  • #5
Yes I recall now. Thanks!
 

Related to Eigenvalue Factorization and Matrix Substitution

1. What is the purpose of eigenvalue factorization?

Eigenvalue factorization is a mathematical technique used to decompose a matrix into its constituent parts, namely eigenvalues and eigenvectors. This allows for easier manipulation and analysis of complex matrices, and is often used in fields such as physics, engineering, and computer science.

2. How is eigenvalue factorization performed?

Eigenvalue factorization involves finding the eigenvectors and eigenvalues of a given matrix through a series of calculations. This is typically done using specialized algorithms, such as the power method or the QR decomposition method.

3. What is the significance of eigenvalues and eigenvectors in matrix factorization?

Eigenvalues and eigenvectors play a crucial role in eigenvalue factorization as they represent the fundamental building blocks of a matrix. Eigenvalues determine the scaling factor of the corresponding eigenvector, and both are used to express the original matrix in a simplified form.

4. What is matrix substitution and how is it related to eigenvalue factorization?

Matrix substitution involves replacing one matrix with another that has the same dimensions and equivalent properties. In eigenvalue factorization, matrix substitution is used to simplify a matrix by replacing it with its eigenvalue decomposition, which can be more easily manipulated or solved.

5. What are some practical applications of eigenvalue factorization and matrix substitution?

Eigenvalue factorization and matrix substitution have a wide range of practical applications, including image and signal processing, data compression, and solving differential equations. They are also commonly used in machine learning algorithms and in the analysis of complex systems, such as financial markets and networks.

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