Math and science::Algebra
LU decomposition
Let \( A \) be a 3x3 square invertible matrix. Elimination on \( A \) by a sequence of elimination matrices \( E_{21}, E_{31} \) and \( E_{32} \) results in an upper triangular matrix \( U \):
\[
E_{32} E_{31} E_{21} A = U
\]
We can combine the elimination matrices together:
\[
E = E_{32} E_{31} E_{21}
\]
and invert to arrive at the LU decomposition:
\[
A = E_{21}^{-1} E_{31}^{-1} E_{32}^{-1} U =E^{-1} U = LU
\]
The elements of the matrix \( L \) have a very direct interpretation, arguably more interpretable than those of \( E \). If \( l_{21}, l_{31} \) and \( l_{32} \) are the scalars by which the first and second rows are subtracted from the rows below, then we can express \( L \) as:
[\[
L = \begin{bmatrix}
? & 0 & 0 \\
? & 1 & 0 \\
? & ? & 1 \\
\end{bmatrix}
\]]