Let $A, B$ be two square matrices with real entries. We say that $A$ and $B$ are similar if an invertible matrix $P$ exists such that $A = P^{-1}BP$.
My question is: if we consider the row echelon form of $A$, it is similar to $A$ itself?
I think that there are some cases in which it could be true (I think about matrices with $LDU$ factorization), but in general it is not true.
Could anyone please help me to understand if it is true or not, and present some example/counterexamples?
The result of the QR algorithm is a matrix that is similar to the original one and in row-echelon form, at least if all eigenvalues are real. Conversely, if $P^{-1}AP$ is upper triangular, then the diagonal elements are the eigenvalues, so the computation of this form can not be simpler than any eigenvalue algorithm, so LU, QR or any other simple decompositions will not give such a result.
The best you get with a simple, non-iterative algorithm is the Hessenberg form of the matrix, which has non-zero elements only above the first sub-diagonal. This form is also the recommended starting point for the QR algorithm and its variants.