Assume we have matrices ${\bf H_i}$ for $i\in[1:K]$ and that the Singluar Value Decomposition (SVD) of ${\bf H_i}$ is such that
$${\bf H_i = A_{bi} D_iA_{si}^*}$$
where ${\bf A_{bi}}$ and $ {\bf A_{si}}$ are vandermonde matrices and are the matrices of the left and right singular vector (described below) and ${\bf D_i}$ is a diagonal matrix .
More precisely
${\bf A_{bi}}= \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & 1 & 1 & \\ e^{j\phi_{1i}} & e^{j\phi_{2i}} & e^{j\phi_{3i}} &\\ e^{j2\phi_{1i}} & e^{j2\phi_{2i}} & e^{j2\phi_{3i}} \\ e^{j3\phi_{1i}} & e^{j3\phi_{2i}} & e^{j3\phi_{3i}} & \\ e^{j4\phi_{1i}} & e^{j4\phi_{2i}} & e^{j4\phi_{3i}} & \\ e^{j5\phi_{1i}} & e^{j5\phi_{2i}} & e^{j5\phi_{3i}} & \\ \end{bmatrix}$ ${\bf A_{si}}= \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & 1 & 1 & \\ e^{j\theta_{1i}} & e^{j\theta_{2i}} & e^{j\theta_{3i}} &\\ e^{j2\theta_{1i}} & e^{j2\theta_{2i}} & e^{j2\theta_{3i}} \\ e^{j3\theta_{1i}} & e^{j3\theta_{2i}} & e^{j3\theta_{3i}} & \\ e^{j4\theta_{1i}} & e^{j4\theta_{2i}} & e^{j4\theta_{3i}} & \\ e^{j5\theta_{1i}} & e^{j5\theta_{2i}} & e^{j5\theta_{3i}} & \\ \end{bmatrix}$
So basically the above matrices have entries $[\bf A_{is}]$ and $[\bf A_{ib}]$ that are complex exponentials (phase shifts) with magnitude equal to one.
My question is next, assume we have formed a concatenated matrix $${\bf H= [{\bf H_1},{\bf H_2},{\bf H_3}, \cdots, {\bf H_K}]}$$ using the matrces ${\bf H_i}$ described above.
Is it safe to say that the right and left singluar values of ${\bf H}$ are also vandermonde matrices and are phase shifts (exponentials) where the entries are of constant magnitude equal to one ..?
Any hints for such a proof if this is true
Thanks!