Getting a unitary matrix and the corresponding triangular matrix

Question

I need to take a matrix $A$ and find a unitary matrix $U$ such that $U^{-1}AU = T$ where $T$ is triangular. The problem I am having is I thought that unitary matrices are ones that deal with complex numbers, however my matrix $A$ is real numbers. Am I missing something? For example if my $A$ was [5 -3; 4 -2] I can also represent that as [5+0i -3+0i; 4+0i -2+0i], but would that still be just the same diagonalizing the matrix in the real numbers, so I won't end up with a triangular matrix $T$. I would end up with a diagonal matrix $D$.

Answer 1

There seems a little confusion in what you have written. A square matrix with real entries can have complex eigenvalues, for example. So you need to ask yourself, before you start solving your problem, which field you are working over. For example, consider

$[ -1 \space{} -2; 2 \space{} 1]$. This matrix has two complex eigenvalues.

Over the complex numbers, every real/complex square matrix is similar to a triangular matrix (Jordan Normal Form). On the other hand, not every square matrix, real or complex, is diagonalisable (whether you consider the solutions over reals or complex numbers).

And then, as you point out, an orthogonal matrix is always also unitary (converse is not true and easy examples are available).