QR decomposition invariance Is the QR decomposition, specifically the $Q$ matrix (I don't care about $R$), invariant to standardization of the matrix being decomposed (i.e. to column-wise zero-mean unit-variance), and if so, what other similar operations is $Q$ invariant to?
 A: It's quite easy to come up with a $2 \times 2$ counterexample. 
The QR factorization of $A = \begin{bmatrix}2&1\\0&2\end{bmatrix}$ is $Q = \begin{bmatrix}1&0\\0&1\end{bmatrix}$ and $R = \begin{bmatrix}2&1\\0&2\end{bmatrix}$. The standardization $A' = \begin{bmatrix}1&-1\\-1&1\end{bmatrix}$ has a QR factorization of $Q' = \begin{bmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$ and $R' = \begin{bmatrix}\sqrt{2}&0\\0&\sqrt{2}\end{bmatrix}$. 
Therefore, the $Q$ matrix is not invariant to standardization of the matrix $A$. 

It is easy to see that for any $n \times n$ matrix $A$ and any $n \times n$ upper-triangular matrix $T$ the matrices $A$ and $A' = AT$ can be factored such that they have same $Q$ matrix. 
Specifically, if $A = QR$ where $Q$ is orthogonal and $R$ is upper-triangular, then $A' = AT = (QR)T = Q(RT)$ where $Q$ is orthogonal and $RT$ is upper-triangular (since $RT$ is the product of two upper triangular matrices). Therefore, multiplying a matrix on the right by an upper-triangular matrix is an operation that won't change the $Q$ matrix. 
