Eliminating one diagonal block produces a smaller matrix governing determinants, inverses, and positivity. A small computation will anchor the general statement before the abstraction takes over.
Definitions first
Matrix factorizations expose different geometry: \(A=QR\) separates an orthonormal frame, while \(A=U\Sigma V^\ast\) separates rotations from axis scaling.
A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.
A small case in full
Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.
The reusable statement
The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.
A nearby false statement
Conditioning matters numerically. An exact algebraic identity can be a poor computational method when it squares the condition number.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.