It can be shown that if a matrix is a triangular matrix then the eigenvalues can be read off as the entries along the diagonal. Does this mean that given any matrix, we can row reduce it to triangular form and read off the eigenvalues along the diagonal? This is demonstrably not (always) true; it seems row reduction distorts information about a matrix's eigenvalues, but it sure got me curious.
Suppose the given matrix, $A$, initially was written in triangular form; in that case we could readily read off the eigenvalues along the diagonal. But that very matrix, $A$, obviously is row equivalent to another non-triangular matrix, $B$, so shouldn't it be possible to row reduce any given matrix to triangular form and read off the eigenvalues?
Best Answer
Row reduction does not preserve the eigenvalues. You can see it by noticing that any invertible matrix (with any eigenvalues of your choice_ is row-reducible to the identity matrix (with all eigenvalues equal to one). Or you can notice that row reduction is achieved by left-multiplying be certain invertible matrices (those realizing the elementary operations); left multiplication by a matrix does not usually preserve eigenvalues.
Eigenvalues are preserved, on the other hand, by similarity (conjugation by an invertible matrix and its inverse). That's why you can always read the eigenvalues from the diagonal of the Jordan form.