Eigenvalues of a row-reduced matrix on the main diagonal still?

It is widely known that if a matrix is given in upper triangular form, then one can just read off the eigenvalues (and their algebraic multiplicity) on the main diagonal of the matrix.

My question is: what if I get a non-upper triangular matrix to start, and I then put it into row-echelon form - not the row-reduced echelon form with all 1's in the pivot variables. Can I spot any of the eigenvalues of the original matrix from this upper triangular matrix?

Thanks,

Solution 1:

$\begin{bmatrix}1&-1\\1&3\end{bmatrix}$ has eigenvalues $\lambda_1 = 2$ and $\lambda_2 = 2$.

$\begin{bmatrix}1&-1\\0&4\end{bmatrix}$ has eigenvalues $\lambda_1 = 4$ and $\lambda_2 = 1$

On the other hand, note that $2\times 2 = 4 \times 1$.

Solution 2:

In general, the eigenvalue preserving transformations are precisely the similarity transformations: $A$ and $B$ have the same eigenvalues precisely when $B=P A P^{-1}$ for some invertible matrix $P$. It is routine to check that elementary row operations are not similarity transformations. For instance, if $EA$ is $A$ after a row operation, then $(EA)^2 \neq EA^2$, in general, yet $(PAP^{-1})^2 = P A^2 P^{-1}$ independent of $P$.

Alternately one can simply do a direct calculation on a $2 \times 2$ example, as in Michael's answer.