Why Do Matrix Expressions Often Involve A A^T in Factorization?

In summary, matrix square factorization is a process of breaking down a square matrix into smaller matrices for easier manipulation and analysis. There are different methods of matrix square factorization, each with its own advantages and best suited for different types of matrices. Using matrix square factorization can simplify complex matrix operations and reveal important properties of the matrix. It can only be applied to square matrices, but there are other methods for non-square matrices. However, it can be computationally expensive for large matrices and not all matrices can be factorized using these methods.
  • #1
MikeLowri123
11
0
Hi All,

I often see this term when factorizing out a matrix from brackets

A(some other term)A^T

where I assume A A^T represents the square within the bracket term, can someone explain the reasoning behind expressions of this kind or point me in the correct direction

Many thanks
 
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  • #2
Can you give a concrete example?
 
  • #3
For example when removing the L term from the variance in the attached equations, can you make this out?
 

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FAQ: Why Do Matrix Expressions Often Involve A A^T in Factorization?

What is matrix square factorization?

Matrix square factorization is a process of decomposing a square matrix into two or more matrices that when multiplied together, result in the original matrix. This is useful in various mathematical and computational applications such as solving linear systems of equations, finding eigenvalues and eigenvectors, and reducing the complexity of matrix operations.

What are the different methods of matrix square factorization?

There are several methods of matrix square factorization, including LU decomposition, QR decomposition, Cholesky decomposition, and Singular Value Decomposition (SVD). Each method has its own advantages and is suitable for different types of matrices.

What are the benefits of using matrix square factorization?

One of the main benefits of using matrix square factorization is that it can simplify complex matrix operations, making them easier to solve. It can also help identify the properties and characteristics of a matrix, such as its rank and determinant, which can be useful in various applications.

Can matrix square factorization be applied to non-square matrices?

No, matrix square factorization can only be applied to square matrices. However, there are other methods for factorizing non-square matrices, such as LU decomposition with pivoting and QR decomposition with pivoting.

Are there any limitations to matrix square factorization?

One limitation of matrix square factorization is that it can be computationally expensive for large matrices. Additionally, not all matrices can be factorized using these methods, particularly those with complex or non-real elements.

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