Joins of Circulant Matrices

The paper proves that for a block matrix with circulant diagonal blocks and constant all-ones off-diagonal blocks, the spectrum is the multiset union of (i) all non-row-sum eigenvalues from the circulant blocks and (ii) the spectrum of an explicit d×d “condensed” matrix built from the row sums and k_j-scaled off-diagonal weights; it also shows how generalized eigenvectors lift and that diagonalizability is equivalent to that of the condensed matrix (main theorem and Section 3) . The candidate solution constructs the same U⊕W invariant decomposition, identifies the same d×d matrix on U, recovers the circulant-block eigenvectors on W, lifts generalized eigenvectors via the natural isomorphism, and concludes the same diagonalizability equivalence. This matches the paper’s Propositions 10, 14 and Corollary 16 in substance, differing mainly in notation and presentation rather than content .