Exercise 1.8.23

The $r$ orthonormal vectors $u_1$ to $u_r$ have $(m-1)+(m-2)+\cdots +(m-r)=\mathit{mr}-\frac{r(r+1)}{2}$ parameters because it takes $m-1$ parameters to specify $u_1$ direction with length 1, and $m-2$ directions to specify $u_2$ direction which is perpendicular to $u_1$, …, and $m-r$ parameters to specify $u_r$ direction which is orthogonal to all previous directions. It is also because there are $r$ constraints that the length of each $u_i$ need to be 1, and $\left(\genfrac{}{}{0ex}{}{r}{2}\right)$ constraints that the inner product of $u_i$ and $u_j$ are zero. So in total we have $\mathit{mr}-r-\frac{r(r-1)}{2}=\mathit{mr}-\frac{r(r+1)}{2}$. And $v_1$ to $v_r$ have $(n-1)+(n-2)+\cdots +(n-r)=\mathit{nr}-\frac{r(r+1)}{2}$ parameters , adding up with the $r$ parameters in $\mathrm{\Sigma }$, we have total $r(m+n-r)$ parameters.

We can also consider $A=\mathit{CR}=(m\times r)(r\times n)$. The matrix $R$ contains an $r$ by $r$ identity matrix, removing $r^2$ parameters from $\mathit{rm}+\mathit{rn}$.