expressing a matrix as a product of 2 or more matrices
Factoring a matrix into 2 matrix L and U
The first question is why the heck do we need LU Decomposition ?
Think of this problem - we need to find $x$ from $Ax = b$
From what we know so far , we can solve this problem by calculating $A^{-1}$ using the method we learnt earlier (using elimination matrix row reduction)
$$ [A \ \ b] = [I \ \ A^{-1}b] = [I \ \ x] $$
But , as we will see now , factorizing the matrix to L and U gives us the opportunity to solve this kind of problems more efficiently by utilizing the properties of L and U
$$ Ax = b \\ LUx = b \\ L(Ux) = b \\ Ly = b $$
So , we can solve for $x$ by solving this pair of equations