# Diagonalization: Eigenvalues Vs Elementary Row Operations [duplicate]

Using elementary row operations, a matrix A $\in \mathrm{R}^{n \times n}$ can be reduced to a Row-Reduced Echelon (RRE) form. Using the RRE form of A, the bases of Nullspace and Range can be obtained. The RRE form of A is a triangular (almost diagonal) form of A and is row-equivalent to A. If we are able to get a simple form of A, why do we need to get eigenvalues and diagonalize A ? What is the motivation for having an alternative method to simplify(diagonalize) A ? Why would one choose to diagonalize A using eigen values ?

## marked as duplicate by kjetil b halvorsen, David K, Daniel W. Farlow, graydad, egregApr 14 '15 at 16:26

Some properties of $A$ cannot be deduced from a row decomposition $A = LR$, where $R$ is the RRE-form of $A$. For example, if you want to compute powers of $A$, for example to compute explicit forms of linear recursion equations, you cannot do this easily from a $LR$-decomposition, we have $$A^n = LR\cdot LR \cdots LR$$ but this is in general not simpler as computing $A^n$ by hand. On the other side, in an eigenvalue decomposition $A = P\Lambda P^{-1}$, $\Lambda$ a diagonal matrix, we have $$A^n = P\Lambda P^{-1} \cdot P\Lambda P^{-1} \cdots P\Lambda P^{-1} = P\Lambda^n P^{-1}$$ and $\Lambda^n$ is computed easily (just take the powers of the diagonal). So some things can be computed more easy with the eigenvalues.