# Conceptual query for finding eigen values during change of basis

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

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).

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

• Thank you. What kind of operations necessarily imply change of basis? – MathMan Jan 24 '14 at 13:57