Is \begin{bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 4 & 4 \\ 0 & 0 & 4 & 4 \end{bmatrix}
an upper triangular matrix? My linear algebra teacher says that the main diagonals must have exclusively pivots or zeros, but I thought that the only requirement for upper triangular form is to have zeros below the main diagonal. Online sources like proofwiki seem to agree with me.
Whose definition is correct?
On
You can have LU factorization of a non-square matrix, where the U is a non-square matrix. see How can LU factorization be used in non-square matrix?
One source that I have has a definition (kind of hidden away in the questions): "An $m\times n$ matrix $A$ is called upper triangular if all entries lying below the diagonal entries are zero, that is, if $A_{ij}=0$ whenever $i>j$." (p.21 Friedberg et al, Linear Algebra 4th edition)
I have yet to find a source that explicitly contradicts this definition (so deliberately states that $m \times n$ matrices cannot be upper triangular), thereby limiting upper triangular matrices to square matrices only. But in all my other sources we have something similar to "...$A \in M_{n \times n}(K)$...upper triangular iff...". The other sources I could consult here was p.37 Cullen (Matrices and linear transformations) and p.149 Golan (The linear algebra a beginning graduate student ought to know).