I am interested on the link between a matrix and and its transformation via a matrix $P=\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 1 & 0\end{bmatrix}$ but I have been unable to find the name of such a matrix $P$ or any interesting litterature about it.
By transformation I mean either going from a matrix $A$ to $A'=PAP^T$:
$A=\begin{bmatrix}a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4}\\a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4}\\a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4}\\a_{4,1} & a_{4,2} & a_{4,3} & a_{4,4}\end{bmatrix}$ and $A'=\begin{bmatrix}a_{1,1} & a_{1,3} & \\a_{3,1} & a_{3,3} \end{bmatrix}$ .
Or going from a matrix $B'$ to $=P^T B'P$:
$B'=\begin{bmatrix}b_{1,1} & b_{1,2} & \\b_{2,1} & b_{2,2} \end{bmatrix}$ and $B=\begin{bmatrix}b_{1,1} & 0 &b_{2,2} & 0 \\0 & 0 &0 & 0 \\b_{1,1} & 0 &b_{2,2} & 0 \\0 & 0 &0 & 0 \end{bmatrix}$.
I wanted to know if there are some keywords that would help me to find some already existing papers/data on the subject ? What would be interesting for me would be to find some kind of relation on the eigenvectors and eigenvalues of $A'$ (resp $B$) knowing those of $A$ (resp $B'$), but any paper on these matrices would already be interesting.
I have some assumptions on $A$ and $B$: they are almost-Toeplitz matrices; symmetric and positive definite (also they are extremely large). The only thing I've been able to show, up to now, is that the non-zero eigenvalues of $B$ are the eigenvalues of $B'$. If somebody have another idea I am interested !
Edit: turns out the eigenvectors of $B$ can be found by adding zeros between components of the eigenvectors of $B'$. The $A\rightarrow A'$ problem seems more complex though.