TITLE: Matrix with orthogonal columns?
QUESTION [6 upvotes]: What do we call a matrix whose columns are orthogonal, such as $\begin{bmatrix}3 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 1 & 0\end{bmatrix}$?
Is there a special name for it? I tried searching to no avail.

REPLY [3 votes]: There is no specific name for it, though such matrices are sometimes called scaled orthonormal/orthogonal matrices. An orthonormal/orthogonal matrix, say $Q$, is a square matrix that satisfies $$Q^*Q = I$$
where $Q^*$ is the conjugate-transpose. The matrices you consider are of the form
$$DQ \text{ or } QD$$ where $D$ is a diagonal matrix.

REPLY [2 votes]: Well, if the columns are orthonormal (i.e. norm 1), then the matrix is orthogonal, and has many beautiful properties.
If not, see Name for matrices with orthogonal (not necessarily orthonormal) rows.
I suppose the right way to think about it is that this matrix maps the standard basis vectors to an orthogonal basis. So it isnt quite preserving distance and angles, but its not too bad.