isHermitian_mul_conjTranspose_self

English

For any A, the product A A^* is Hermitian.

Русский

Для любой матрицы A произведение A A^* является Гермитовой.

LaTeX

$$$(A^{\\mathrm{H}} A)^{\\mathrm{H}} = A^{\\mathrm{H}} A$$$

Lean4

/-- Note this is more general than `IsSelfAdjoint.mul_star_self` as `B` can be rectangular. -/
theorem isHermitian_mul_conjTranspose_self [Fintype n] (A : Matrix m n α) : (A * Aᴴ).IsHermitian := by
  rw [IsHermitian, conjTranspose_mul, conjTranspose_conjTranspose]

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