Commit 0cd2885c authored by D'azevedo, Ed's avatar D'azevedo, Ed

minor update

parent 4d408ae1
......@@ -249,18 +249,20 @@ block diagonal matrix of the form
$\tV_k \otimes \tU_k$ (such as moving the center site from
site $i$ to site $i+1$), then we may have
the change of basis operation by evaluting
$ \mbox{pinv}(V_k \otimes U_k) * (\tV_k \otimes \tU_k)$
$ \mbox{pinv}(\tV_k \otimes \tU_k) * (V_k \otimes U_k) * \mbox{vec}(D_k)$
where $\mbox{pinv}$ is the least-squares pseudo-inverse.
%
Since $V_k$ and $U_k$ have ortho-normal columns,
then $\mbox{pinv}(V_k) = V_k'$
and $\mbox{pinv}(U_k) = U_k'$ and
$\mbox{pinv}( V_k \otimes U_k) = V_k' \otimes U_k'$.
Since $\tV_k$ and $\tU_k$ have ortho-normal columns,
then $\mbox{pinv}(\tV_k) = \tV_k'$
and $\mbox{pinv}(\tU_k) = \tU_k'$ and
$\mbox{pinv}( \tV_k \otimes \tU_k) = \tV_k' \otimes \tU_k'$.
%
Then $(V_k' \otimes U_k') * (\tV_k \otimes \tU_k)$ can be
evaluated as $(V_k'*\tV_k) \otimes (U_k' * \tU_k)$.
Then $(\tV_k' \otimes \tU_k') * (V_k \otimes U_k)* \mbox{vec}(D_k)$ can be
evaluated as $(\tV_k'*V_k) \otimes (\tU_k' * U_k)* \mbox{vec}(D_k)$ or
$(\tU_k' * U_k) * (D_k) * ( \tV_k' \otimes V_k')^t$.
%
Note that in general the rectangular matrices $V_k$ and $\tV_k$
Note that in general the rectangular matrices
$V_k$, $\tV_k$, $U_k$, $\tU_k$
have in compatible shapes and one may assume the matrices
are padded with zeros to be of the appropriate size.
%
......
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment