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I have a banded bisymmetric matrix, say

$ A = \begin{pmatrix} a & d & f & 0 & 0 \\ d & b & e & g & 0 \\ f & e & c & e & f \\ 0 & g & e & b & d \\ 0 & 0 & f & d & a \end{pmatrix} $,

not necessarily a tridiagonal 5x5 matrix, but a general bisymmetric NxN with band K.

While I know that the inverse of a band-limited matrix is not necessarily banded, can we say something if we have the above symmetry in addition ? Like, is it banded and, if so, what is the length of band ?

If the inverse is still not necessarily band-limited, is there any additional property that could help ?

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    $\begingroup$ A single randomly-generated example in MATLAB shows that this extra symmetry does not ensure that the inverse of a banded matrix is banded. $\endgroup$ Apr 2 at 23:22
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    $\begingroup$ $\pmatrix{2&1&0\cr1&3&1\cr0&1&4\cr}^{-1}=(1/18)\pmatrix{11&-4&1\cr-4&8&-2\cr1&-2&5\cr}$ $\endgroup$ Apr 2 at 23:27

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