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Matrix inversion preserves regularity where the determinant is nonzero
Statement
Let , let be open, and let . If the entries of are and never vanishes, then the entries of are .
Facts & Assumptions
Given: The matrix-valued map in the Statement. Cofactors have the meaning of Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring, and the reciprocal derivative rule is supplied by Sums, scalar multiples, products and quotients: , , , and when .
For an invertible square matrix, (If is a unit, then )
The function is evaluation of the polynomial (For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries)
Finite componentwise sums and products of Euclidean maps are , and a composite of composable Euclidean maps is ( Euclidean maps are closed under componentwise algebra and composition).
Proof
Every cofactor and the determinant are polynomial expressions in the entries of , so they are by [L2] and [L3]; [L1] identifies the only remaining factor needed for the inverse.
On , repeated differentiation of gives . This formula follows by induction from the reciprocal and product rules, and every derivative displayed is continuous on that domain.
Since never vanishes, its image lies in the domain of step 1.2. Thus is by composition, and [L1] together with step 1.1 and [L3] makes every entry of .
Depends on
- $C^k$ Euclidean maps and diffeomorphisms
- $C^k$ Euclidean maps are closed under componentwise algebra and composition
- If $\det(A)$ is a unit, then $A^{-1}=\det(A)^{-1}\operatorname{adj}(A)$
- Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring
- For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
Used by
- Connection on a smooth vector bundle Definition
- Choice-free smooth inverse function theorem in Euclidean space Lemma
- Cotangent coordinate changes use the inverse transpose Jacobian Lemma
- Tangent-bundle chart transitions are smooth with smooth inverses Lemma
- A fibrewise bijective smooth bundle map over a diffeomorphism is a bundle isomorphism Proposition
- Constant-rank kernels and images of bundle maps over one base are subbundles Proposition
- A local inverse of a Cᵏ regular map is Cᵏ Theorem
- Dual and Hom transition functions define smooth bundles Theorem
- The parametrized implicit function theorem with Cᵏ regularity Theorem
Dependency tree · two levels
34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis II, §§8.5–8.6 (standard reference, not scraped)