Conceptual query for finding eigen values during change of basis

Question

Consider an n x n matrix. Suppose i wish to find the eigen values of this matrix.

Now, we know that row transformation is equivalent to a change of basis in the vector space. But, we also know that a change of basis preserves the eigen value of a matrix.

This means, if i am able to change the given matrix into an upper triangular matrix using row/column transformations, the eigen values should remain intact under change of basis.

This means, the eigen values should be found on the diagonal of the triangular matrix obtained , which is not the case. Where could the method go wrong?

Thanks

Answer 1

As pointed out in the other answer, row operations (or column operations) need not preserve eigenvalues. What you need are similarity transformations:

$$A\mapsto P^{-1}AP$$

for some invertible $P$; this is what corresponds to a change of basis.

More conceptually, when you have a linear map $f\colon V\to W$, row operations correspond to a change of basis of either $V$ or $W$ (and column operations to the other). Which one is which depends on some conventions.

But when you're talking about eigenvalues, you have a map $f\colon V\to V$, and you should pick a single basis for $V$, so you can't change the bases of the domain and codomain independently. So any row operation you do has to come with a corresponding column operation (corresponding to multiplying by the inverse of the elementary matrix on the other side).

Answer 2

Row operations do NOT correspond to a change of basis, and will very likely affect the eigenvalues.