I am reading about matrices with orthogonal columns (not necessarily an orthogonal matrix). Wikipedia says:
It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy $M^TM = D$, with $D$ a diagonal matrix.
Is it also true that $MM^T = D$ implies that $M$ has orthogonal columns?
No, $MM^T = D$ only implies that $M$ has orthogonal rows.
For example $$ M = \begin{pmatrix}1 & 2 \\ -6 & 3\end{pmatrix} $$ yields $$ MM^T = \begin{pmatrix} 5 & 0 \\ 0 & 45\end{pmatrix} $$ and the rows are indeed orthogonal. However $$ M^T M = \begin{pmatrix} 37 & -16 \\ -16 & 13\end{pmatrix}, $$ so the columns are not orthogonal.