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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?

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Yes, it is correct, the Gauss elimination changes the basis.

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.

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