# What is known about merely-orthogonal matrices?

I'm interested in square matrices whose columns are orthogonal, but not necessarily orthonormal, non-zero vectors. Answers to other questions on this topic have noted that such matrices do not have an agreed name and that the natural name of "orthogonal matrix" means a matrix with orthonormal columns. So I'm going to use the name "merely-orthogonal" for these matrices and "orthonormal" for those matrices where $$Q^TQ=I$$, avoiding "orthogonal matrix" entirely.

The answer to one question usefully notes that if $$M$$ is merely-orthogonal then there exists an invertible diagonal matrix $$D$$ and an orthonormal matrix $$Q$$ such that $$M=QD$$. The elements of $$D$$ are easily found as the norms of each column. This also immediately implies that $$M^TM = (QD)^TQD = D^TQ^TQD = D^T D = D^2$$ which is diagonal and has non-zeros on the diagonal. And hence, $$M$$ is invertible with $$M^{-1} = D^{-2}M^T$$ It seems like merely-orthogonal matrices should form a group under multiplication and that there should be an analog to the $$QR$$ decomposition (call it the $$MR$$ decomposition) where for any matrix $$A$$, $$A = MR$$ with $$M$$ merely-orthogonal and $$R$$ upper triangular. It seems to me that the $$MR$$ decomposition would not require the underlying field to be algebraically closed, like the $$QR$$ decomposition does but would be computable over the rational numbers.

So my questions are:

1. Has anyone studied merely-orthogonal matrices and established these or other results?
2. Am I right that merely-orthogonal matrices form a group?
3. Am I right that $$MR$$ decompositions can be done over the rationals?

For example, all invertible diagonal matrices are merely orthogonal, but $$\begin{pmatrix} 1&-1\\ a&b \end{pmatrix}$$ is merely orthogonal iff $$ab=1$$, which is not preserved under the multiplied by $$\operatorname{diag}(1,\lambda)$$ for $$\lambda\neq\pm 1$$ scaling the second row. Also, the matrix $$\begin{pmatrix} \cos\theta&-R\sin\theta\\ \sin\theta&R\cos\theta \end{pmatrix}$$ is merely orthogonal but its inverse $$\begin{pmatrix} R\cos\theta&R\sin\theta\\ -\sin\theta&\cos\theta \end{pmatrix}$$ is not unless $$R=\pm 1$$.
There are of course the conformal linear transformations, which preserve conformal frames (i.e., orthogonal basis all of the same length) and they form a group -- the conformal group (fixing $$0$$) consisting of (nonzero) scalar multiples of orthogonal matrices.
• Thank you. The observation that the merely-orthogonal matrices are not closed under multiplication basically answers all the questions. 1. Nobody studies them because they aren't that useful 2. No they don't form a group 3. The $MR$ decomposition would be very hard to construct since you can't form $M$ as the product of simpler merely-orthogonal matrices like you form $Q$ from Householder reflections or Givens rotations Commented Oct 19, 2023 at 18:14