Solving Linear Systems with LU Decomposition and complete pivoting; stupid question Given a matrix A and vector B, solve $Ax=B$ Using LU Decomposition with full Pivoting; $PAQ=LU$ where P and Q are row and column permutation vectors (correct me if I'm wrong) What I don't understand is what to do with the permutation matrices to finish the solution. I know in partial pivoting, its simple $Lz=PB$ $Ux=z$ But what do I do with Q? PS If anyone is a C head, you're help would be appreciated in the implementation

Recall that permutation matrices have the property that $P^{-1} = P^T$, so we can re-arrange the factorization to write $A$ in the form $A = P^TLUQ^T$. After that, it is straightforward to solve:

$\begin{align} Lz &= Pb \\\ Uy &= z \\\ x &= Qy \\\ \end{align}$

Note that $Q$ can't be just "ignored" because it is a "column permutation". It is a column permutation by virtue of how it is used (right-multiplication), not by virtue of the structure of the matrix, and it _does_ have an effect when applied to a column vector with left-multiplication.