11 - Row Rank Criterion on Product of Matrices

Row Rank Criterion on Product of Matrices

Let $P=QR$ where $P,Q,R$ are respectively $k\times l,k\times m,m\times l$ matrices over $\mathbf{F}$

1. $\mathrm{rowrank}(P)$ is at most $m$

2. Let $p=\mathrm{rowrank}(P)$, then $P$ can be written as a $k\times p$ and $p\times l$ matrix

3. The row rank and column rank of a matrix is equal

Proof

1. By Property of Row Space we have

$$\mathrm{Row}(P)=\mathrm{Row}(QR)\le \mathrm{Row}(R)$$

Hence

$$\mathrm{rowrank}(P)=\dim \mathrm{Row}(P)\le \dim \mathrm{Row}(R)=\mathrm{rowrank}(R)\le m$$

2. There is a $k\times k$ invertible matrix $E$ such that $EP=\mathrm{RRE}(P)$ and hence

$$P=E^{-1}\mathrm{RRE}(P)$$

Let $\tilde{E}$ denote the first $p$ columns of $E^{-1}$ and $\tilde{P}$ denote the first $p$ rows of $\mathrm{RRE}(P)$
As the last $k-p$ rows of $\mathrm{RRE}(P)$ are zero rows then

$$P=\tilde{E}\tilde{P}$$

where $\tilde{E}$ is a $k\times p$ and $\tilde{P}$ is a $p\times l$ matrix
3) From $(1)$ and $(2)$, the row rank of a $k\times l$ matrix $P$ is the minimal value $p$ such that
$P$ can be written as the product $QR$ of a $k\times p$ matrix and a $p\times l$ matrix
Whenever $P=QR$

$$\underset{l\times k}{P^T}=\underset{l\times p}{R^T}\underset{p\times k}{Q^T}$$

So the row rank of $P^T$ is similarly $p$. But the $\mathrm{rowrank}(P^T)=\mathrm{colrank}(P)$ as required