suppose I have a matrix $A\_{1}\in C^{m\*n}, m\geqslant n$ so the QR decomposition of $A\_{1}$ is $A\_{1}=Q\_{1}\*R\_{1}$. Now, define the augmented matrix $A\_{2}=\begin{bmatrix}
A\_{1}\\ 
I\_{n\*n}
\end{bmatrix}$ where $I\_{n\*n}$ is $n\*n$ identity matrix, so the QR decomposition of $A\_{2}$ is $A\_{2}=Q\_{2}\*R\_{2}$.  
 Is there any relation between $Q\_{1}$ and $Q\_{2}$, $R\_{1}$ and $R\_{2}$, and especially between the diagonal elements of $R\_{1}$ and $R\_{2}$ knowing that $det(I\_{n\*n}+A\_{1}^{H}\*A\_{1})=det(A\_{2}^{H}\*A\_{2})$.  




Relation between the QR decomposition of matrix A and the QR of the augmented version of A
First consider the simple case, if $Q\_1 = I$, then you want to find $Q\_2$ and $R\_2$ such that $Q\_2 R\_2 = \begin{bmatrix} R\_1 \\ I \end{bmatrix}$. Split the columns of $Q\_2^T$ into left and right halves such taht $Q\_2^T = [U V]$. Note that $U^TU = V^TV = I$ and $U^TV = 0$. Then basically you want to find $R\_2 = UR\_1 + V$ which implies that $U^TR\_2 = R\_1$. 


The more general case can be analyzed by considering the relation that $Q\_2R\_2 = \begin{bmatrix} Q\_1R\_1 \\ Q\_1Q\_1^T \end{bmatrix}$ and therefore $R\_2 = UQ\_1R\_1 + V$ which leads to the result that $Q\_1^TU^TR\_2 = R\_1$.


