We know that for any complex $(m \times n)$-matrix $A$ with linearly independent columns, there exists an $(m \times n)$-matrix $Q$ and an $(n \times n)$-matrix $R$ such that $A = QR$, $Q^* Q = I$, $R$ is a invertible upper triangular matrix, and the columns of $Q$ are an orthonormal basis for $\operatorname{Im}(A)$.
But I wonder if the columns of $A$ can form a orthogonal basis of $\operatorname{Im}(A)$?
The columns of $A$ are orthogonal if and only if the upper-triangular matrix $R$ is diagonal.