[Math] QR decomposition – does Q always have full column rank

linear algebramatrices

Given $A \in R^{m\times n}$, $Q \in R^{m\times n}$ and $R \in \mathbb{R}^{n\times n}$ with $m \ge n$, does $Q$ always have $n$ independent columns?

I played around in matlab by generating a random $m\times n$ matrix $B$ and setting $A = BB^{T}$. When I do the QR-decomposition, I always get a $Q$ with full rank but $R$ singular. Not sure how to construct a more mathematical argument though.

Best Answer

Note that the matrix $Q$ always has columns to be orthonormal and therefore the matrix $Q$ will always have full-rank. In general, you could have three different $QR$ factorizations; If $A$ is a $m \times n$ matrix with rank $r$ and $m \geq n$, we could then have

  1. Full $QR$: $Q \in \mathbb{R}^{m \times m}$ and $R \in \mathbb{R}^{m \times n}$.
  2. Reduced $QR$: $Q \in \mathbb{R}^{m \times n}$ and $R \in \mathbb{R}^{n \times n}$.
  3. Reduced rank $QR$: $Q \in \mathbb{R}^{m \times r}$ and $R \in \mathbb{R}^{r \times n}$.

In all these cases, the columns of $Q$ are orthonormal, while the matrix $R$ is upper triangular.

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