# How to write block matrices on diagonal in nice form?

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?

$$Z=\begin{bmatrix} a&b\\ c&a+d&b\\ &c&a+d&b\\ &&c&a+d&b\\ &&&\ddots&\ddots&\ddots\\ &&&&c&a+d&b\\ &&&&&c&d \end{bmatrix},$$ 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.
• (first of all, you can symmetrize the matrix, which gives a similar matrix, by keeping the same main diagonal and switch the two off diagonals to $\sqrt{b}\sqrt{c}$) Dec 1, 2019 at 19:47