6. Matrix Decomposition Methods

Matrix decomposition, also known as matrix factorisation, is the process of breaking down a matrix into simpler components that can be more easily analyzed or manipulated. This involves expressing a matrix as a product of two or more matrices, each with their own properties and applications.

For example,

\[\begin{split} A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & -2 \end{pmatrix} \end{split}\]

Here we will be looking at three common matrix decomposition methods:

• LU decomposition

• Cholesky decomposition

• QR decomposition