Take a (kind of) arrowhead real-symmetric matrix of the general form
$$ M = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{12} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{13} & a_{23} & a_{33} & 0 & 0 & 0 \\ a_{14} & a_{24} & 0 & a_{44} & 0 & 0 \\ a_{15} & a_{25} & 0 & 0 & a_{55} & 0 \\ a_{16} & a_{26} & 0 & 0 & 0 & a_{66} \\ \end{bmatrix} $$
where the size of the blocks may vary, however in general, the diagonal submatrix will be of dimension close to that of the entire matrix. Is there a method to diagonalise this matrix which takes advantage of this largely diagonal structure? My desire is computational efficiency, i.e. compared to dgemm.
I require all of the eigenvalues and eigenvectors of this matrix, i.e. $V^{-1}MV = W$ where $V$ are the eigenvectors of $M$, and $W$ a diagonal matrix containing the eigenvalues.
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