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 bellowenter image description here

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.

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  • $\begingroup$ So it has name "corner perturbations", thank you $\endgroup$ – Jan Filip Dec 1 '19 at 19:23
  • $\begingroup$ Yes, "corner perturbations" could also indicate a few more elements in the corners that are changed, obvious by context. More general "low-rank perturbation". If you are interested in the eigenvalues or inversion there are closed formulas for many cases; see for example cambridge.org/core/journals/anziam-journal/article/… $\endgroup$ – whoooo Dec 1 '19 at 19:39
  • $\begingroup$ What I really need is determinant of it. But you already helped me a lot $\endgroup$ – Jan Filip Dec 1 '19 at 19:42
  • $\begingroup$ If you can tell me a,b,c,d then I can tell you if there are closed formulas for the eigenvalues. $\endgroup$ – whoooo Dec 1 '19 at 19:45
  • $\begingroup$ (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 $\sqrt{b}\sqrt{c}$) $\endgroup$ – whoooo Dec 1 '19 at 19:47

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