Why QR-decomposition returns transpose of Q?

Question

In the documentary we read about QRDecomposition (Link)

QRDecomposition[m] yields the QR decomposition for a numerical matrix m. The result is a list {q,r}, where q is a unitary matrix and r is an upper‐triangular matrix.

By QR-decomposition we can decompose the matrix m=q.r. Therefore I was surprised to find following point in the same description:

The original matrix m is equal to ConjugateTranspose[q].r.

What is the reason that MMA returns the conjugate transpose of q but not q? Because of this reason we can decompose matrix m only by the conjugate transpose of q.

MMA 13.2

Answer 1

$q$ is a unitary matrix, so the conjugate transpose is actually the inverse: $q^+=q^{-1}$. In this sense, the equation that is satisfied,

$$ m=q^+\cdot r=q^{-1}\cdot r $$

can be written as

$$ q\cdot m= r $$

which is the form of the equation that is, in practice, often very useful. It answers the question: "What do I need to do to $m$ to turn it into a triangular matrix?"

So I guess they picked this latter form when deciding how to deliver the results, for reasons we can ultimately only guess at.

The same remains true for pivoting: QRDecomposition[m, Pivoting->True] returns $\{q,r,p\}$ that satisfy

$$ q\cdot m\cdot p=r $$ and in this sense the matrices $q$ and $p$ serve to turn $m$ into the triangular matrix $r$.