6. Matrix Decomposition Methods#
Matrix decomposition, also known as matrix factorization, 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, consider the matrix \(A\) below
\[\begin{split} A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix}. \end{split}\]
This can be decomposed into two matrices \(L\) and \(U\) such that \(A = LU\) where
\[\begin{split} \begin{align*}
L &= \begin{pmatrix}
1 & 0 & 0 \\
4 & 1 & 0 \\
7 & 2 & 1
\end{pmatrix}, &
U &= \begin{pmatrix}
1 & 2 & 3 \\
0 & -3 & -6 \\
0 & 0 & 0
\end{pmatrix}.
\end{align*} \end{split}\]
Here we will be looking at three common matrix decomposition methods: