I'm trying to show that for any square matrix (whose entries are all real) the eigenvalues are real or are complex conjugate pairs.
I've tried so far by stating that for 2*2 matrices, finding the determinant of (A-$\lambda$I) always leads to a quadratic which inherently yields real or complex roots of the form of $z=a\pm\ b$ although I doubt that this counts as sufficient proof for those cases anyway.
Am I on the right track at least? Am I also assuming incorrectly that any square matrix beyond 2*2 dimensions has eigenvalues that are either real or complex conjugate pairs?
Best Answer
The eigenvalues are the roots of the characteristic polynomial, and the coefficients of the characteristic polynomial are real since they depends on the element of the matrix. This gives that complex eigenvalues come in conjugate pairs.