Let $Y$ be real $2\times 2$ matrix
\begin{equation*}
Y = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\end{equation*}
and $Z$ is block matrix constructed as depicted on the picture bellow
Matrices $Y$ are lying on diagonal of matrix $Z$ and they are overlapping such that gray areas can be writen as $a + d$.
Is there cleaner definition of matrix $Z$ in terms of $Y$ so one has not to write elements of $Z$ elementwise? Can we use Toeplitz matrix?
So the matrix is
$$ Z=\begin{bmatrix} a&b\\ c&a+d&b\\ &c&a+d&b\\ &&c&a+d&b\\ &&&\ddots&\ddots&\ddots\\ &&&&c&a+d&b\\ &&&&&c&d \end{bmatrix}, $$ which is the Toeplitz matrix $T_n(f)$ generated by $f(\theta)=a+d+ce^{\mathbf{i}\theta}+be^{-\mathbf{i}\theta}$ + the low-rank matrix
$$ \begin{bmatrix} -d\\ \\ \\ \\ &&&&-a \end{bmatrix}. $$ This is a standard scalar-valued tridiagonal Toeplitz matrix plus corner perturbations. No need to do any blocking.