Gram-Schmidt process: dependence on matrix Assume that we are given a real $n\times k$ matrix $A$ ($n\geq k$), of rank $k$. Moreover assume that we are given an invertible $k\times k$ matrix $C$.
We apply the Gram-Schmidt process to the columns of $A$ and $A\cdot C$ to obtain matrices $B$ and $D$ with the property $B^* B = I_k = D^* D$.
What can we say about the relationship between $B$ and $D$?
To be more precise: is there an orthogonal matrix $E\in O_k$ such that $B = D\cdot E$?
Thanks!
 A: So maybe it works like this:
Since $A$ has rank $k$, we get an unique QR-decomposition $A=QR$, where $Q^*Q=I_k$ and $R$ is an upper triangular matrix with positiv diagonal entries, in particular it is invertible, hence $R\in GL_k$. Then if we apply the Gram-Schmidt process to $R$ we get an orthogonal matrix $R_1$. Then we can write $A C= Q R C$, where $R C$ is again invertible and we can apply the Gram-Schmidt process to $R C$ to obtain an orthogonal matrix $R_2$. Hence one could say that transforming $A$ respectively $A C$ to a (rectangular) orthogonal matrix one gets the same up to on orthogonal matrix.
What do you think?
A: The columns of $A\cdot C$ are linear combinations of the columns of $A$ (the coefficients in the linear combinations being the entries of $C$), and vice versa (since $C$ is invertible).  So the columns of $A\cdot C$ span the same linear subspace $S$ as the columns of $A$.  The columns of $B$ constitute an orthonormal base for $S$, and so do the columns of $D$. Any two orthonormal bases for the same space are related by an orthogonal transformation.
