English
There is a natural isomorphism between the type of 1-by-1 matrices over A and A itself, identifying the unique matrix with its single entry.
Русский
Существует естественное изоморование между множеством матриц размером 1×1 над A и A самó по себе, каждая единичная матрица соответствует своему единственному элементу.
LaTeX
$$$M \cong A$ via \; M_11 \mapsto M_{11}$, i.e., 1×1 matrices are canonically isomorphic to the base ring.$$
Lean4
/-- The isomorphism between the type of all one by one matrices and the base type. -/
@[simps]
def uniqueEquiv : Matrix m n A ≃ A where
toFun M := M default default
invFun a := .of fun _ _ => a
left_inv M := by ext i j; simp [Subsingleton.elim i default, Subsingleton.elim j default]
right_inv a := by simp
