isSymm — Mathlib

English

The isDiag property of fromBlocks corresponds to the diagonal blocks and zero off-diagonal blocks: (A.fromBlocks B C D).IsDiag iff A.IsDiag ∧ B = 0 ∧ C = 0 ∧ D.IsDiag.

Русский

Свойство диагональности fromBlocks эквивалентно тому, что A и D диагональны, а B и C нулевые.

LaTeX

$$$\\forall {\\alpha} {n} {m} [Zero \\alpha] {A : Matrix m m \\alpha} {B : Matrix m n \\alpha} {C : Matrix n m \\alpha} {D : Matrix n n \\alpha}, (A.fromBlocks B C D).IsDiag \\iff A.IsDiag ∧ B = 0 ∧ C = 0 ∧ D.IsDiag$$$

Lean4

theorem isSymm [Zero α] {A : Matrix n n α} (h : A.IsDiag) : A.IsSymm :=
  by
  ext i j
  by_cases g : i = j; · rw [g, transpose_apply]
  simp [h g, h (Ne.symm g)]

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