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Nov 3, 2022 at 12:35 comment added Denis Serre @PadraigÓCatháin I don't think so. I don't use the multiplication of blocks. The formula $K=U^*DU$ where $D$ is block-diagonal is always valid. And it is known that $\det D$ is the product of the determinants of the diagonal blocks.
Nov 3, 2022 at 10:45 comment added Padraig Ó Catháin Block-circulant can be used in two senses: that the blocks are circulant, or that the template in which the blocks are placed is circulant. Your formula requires both senses! In general, there are formulas for determinants of a block matrix in terms of polynomial expressions in the blocks provided that the blocks commute.
Nov 2, 2022 at 23:04 vote accept user369335
Nov 2, 2022 at 19:54 history edited Denis Serre CC BY-SA 4.0
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Nov 2, 2022 at 17:03 history answered Denis Serre CC BY-SA 4.0