updateRow_conjTranspose — Mathlib

English

The Kronecker product of the identity on the left with any B is, up to a permutation of indices, the block-diagonal matrix with blocks equal to B.

Русский

Кронекерово произведение I_m ⊗ B равно, после перестановки индексов, блочно-диагональной матрице с блоками B.

LaTeX

$$$$ I_m \\otimes B = \\operatorname{reindex}(\\operatorname{Equiv.prodComm}_, \\operatorname{Equiv.prodComm}_)\\big(\\operatorname{blockDiagonal}(\\lambda _\\mapsto B)\\big) $$$$

Lean4

theorem updateRow_conjTranspose [DecidableEq n] [Star α] : updateRow Mᴴ j (star c) = (updateCol M j c)ᴴ :=
  by
  rw [conjTranspose, conjTranspose, transpose_map, transpose_map, updateRow_transpose, map_updateCol]
  rfl

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