I am looking for a regular tridiagonal matrix $A$ such that at the LU-decomposition with partial column pivoting the matrices $L$ and $U$ are also tridiagonal, but with total pivoting the matrices $L$ and $U$ are no triadiagonal matrices.
I thought of a tridiagonal matrix that is diagonally dominant, but then we would not need to apply partial column pivoting. But what about total pivoting ?
Or is there an other example of a such a matrix $A$ ?