Question on decomposition of a matrix

In summary, the conversation discusses a 2x2 real matrix M that can be decomposed into M=ATΣA, where Σ is symmetric positive definite and A is a nonsingular 2x2 matrix. The speaker is wondering if it is possible to find another matrix PM such that PM=AQ, where Q does not depend on A and the multiplication with A only appears at the left side. It is mentioned that the Cholesky normal form and diagonalization techniques can be used to find such a matrix for 2x2 matrices.
  • #1
mnb96
715
5
Hello,

I have a [itex]2\times 2[/itex] real matrix [itex]M[/itex] such that: [tex]M=A^T \Sigma A[/tex], where the matrix [itex]\Sigma[/itex] is symmetric positive definite, and [itex]A[/itex] is an arbitrary 2x2 nonsingular matrix. Both A and ∑ are unknown, and I only know the entries of the matrix M itself. Note that M is symmetric positive definite too.

I was wondering if it is possible to apply some decomposition of the matrix [itex]M[/itex] in order to find another matrix [itex]P_M[/itex] such that: [tex]P_M = AQ[/tex]
where the matrix Q must not depend on A (e.g. it cannot be a product of matrices where A appears). I basically want to find a matrix PM where the multiplication with A appears only at the left side, and not at both sides like in M.
 
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  • #2
The Cholesky normal form for symmetric positive definite matrices ##M## gives us ##M=LDL^\tau## as yours. But we have additionally that ##D## is diagonal and ##L## a lower triangular matrix. If we take ##D^{1/2}\cdot D^{1/2}=D## the square root of ##D## and set ##S:=LD^{1/2}## we can even write ##M=SS^\tau## with a lower triangular matrix ##S##. For ##2\times 2## matrices, this should give you some nice conditions.
 

1. What is the purpose of decomposing a matrix?

Decomposing a matrix allows us to break down a complex matrix into simpler components, making it easier to analyze and solve problems involving the matrix.

2. What are the different types of matrix decomposition?

The most common types of matrix decomposition are LU decomposition, QR decomposition, Cholesky decomposition, and Singular Value Decomposition (SVD).

3. How is matrix decomposition useful in real-world applications?

Matrix decomposition is used in various fields such as data analysis, engineering, and computer science. It helps in solving systems of linear equations, image processing, and data compression.

4. What is the difference between LU and QR decomposition?

LU decomposition decomposes a matrix into a lower triangular matrix and an upper triangular matrix, while QR decomposition decomposes a matrix into an orthogonal matrix and an upper triangular matrix.

5. What is the process of decomposing a matrix?

The process of decomposing a matrix involves finding a set of matrices that when multiplied together, give the original matrix. This can be done using various methods such as Gaussian elimination, Gram-Schmidt process, or singular value decomposition.

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