# Existence of Upper Triangular Matrix Over Real Numbers

I'm studying Sheldon Axler's Linear Algebra Done Right. In Chapter 5 on Eigenvalues and Eigenvectors, he introduces the upper triangular matrix and proves that over a complex vector space, every operator has an upper-triangular matrix, using the fact that over $$\mathbb{C}$$, every operator has an eigenvalue.

My question is, is this true for vector spaces over $$\mathbb{R}$$?

It seems to me that one can apply Gaussian Elimination on any square matrix and get an upper triangular-matrix and thus any operator on a real vector space has an upper-triangular matrix.

But when I apply Gaussian elimination, it is left multiplying by a matrix, so I am changing the basis of the codomain, is this correct?

Any matrix with a nonreal eigenvalue gives an example, e.g. $$\pmatrix{0&1\\-1&0}$$ can't be written as an upper triangular matrix in any basis, because the equations for the sum and product of the diagonal elements would give the eigenvalues $$\pm i$$ which are not real.