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,
Best Answer
$\begin{bmatrix}1&-1\\1&3\end{bmatrix}$ 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 $2\times 2 = 4 \times 1$.