LU Decomposition Steps I've been looking at some LU Decomposition problems and I understand that making a matrix A reduced to the form A=LU , where L is a lower triangular matrix and U is a upper triangular matrix, however I am having trouble understanding the steps to get to these matrices. Could someone please explain the method for LU Decomposition in detail, preferably excluding the concept of permutation matrices? ( We haven't talked about permutation matrices in class yet so our professor forbids us to use them for the decomposition).
 A: $LU$ decomposition is really just another way to say Gaussian elimination.
If you're familiar with that, putting the pieces together is easy.
Here is an example. Let
$$
A=A^{\left(0\right)}=\left[\begin{array}{ccc}
8 & 1 & 6\\
4 & 9 & 2\\
0 & 5 & 7
\end{array}\right].
$$
Proceed by Gaussian elimination. The first multiplier is $\ell_{2,1}=4/8=0.5$
(this is the multiplier that allows us to cancel $a_{2,1}=4$ using
the first row) and the second is $\ell_{3,1}=0/8=0$.
We arrive at
$$
A^{\left(1\right)}=\left[\begin{array}{ccc}
8 & 1 & 6\\
0 & 8.5 & -1\\
0 & 5 & 7
\end{array}\right].
$$
To cancel out $a_{3,2}^{\left(1\right)}=5$, we use the multiplier
$\ell_{3,2}=5/8.5\approx0.5882$ to yield
$$
A^{\left(2\right)}\approx\left[\begin{array}{ccc}
8 & 1 & 6\\
0 & 8.5 & -1\\
0 & 0 & 7.5882
\end{array}\right]
$$
which yields the $LU$ decomposition
$$
A=LU\approx\left[\begin{array}{ccc}
1 & 0 & 0\\
0.5 & 1 & 0\\
0 & 0.5882 & 1
\end{array}\right]\left[\begin{array}{ccc}
8 & 1 & 6\\
0 & 8.5 & -1\\
0 & 0 & 7.5882
\end{array}\right].
$$
Note that $L$ is just made up of the multipliers we used in Gaussian
elimination with $1$s on the diagonal, while $U$ is just $A^{\left(2\right)}$.
