2.21.2 Tensor Isomorphism Invertibility

Tensor Isomorphism Invertibility is the property, and the collection of equivalent tests for that property, by which a linear map between vector spaces is confirmed to possess a two-sided inverse, together with the consequences of that property for the maps and bilinear forms, such as metric tensors, that tensor algebra induces or depends on. Where the isomorphism linear map entry examines a single map's bijectivity and constructs an example explicitly, invertibility is treated here as a property to be tested, characterized by several equivalent criteria, and traced through to its consequences whenever an invertible map or form is used to build further tensor structure, such as raising and lowering indices.

Equivalent Characterizations of Invertibility

The Core List of Equivalences

For a linear map φ : V → V on a finite-dimensional space, or equivalently for its matrix M in any basis, the following conditions are all equivalent, and any one of them certifies that φ is invertible:

together with: ker(φ) = {0}; the columns of M are linearly independent; the rows of M are linearly independent; φ is surjective; every eigenvalue of φ is nonzero; M can be reduced to the identity matrix by elementary row operations. Each of these is a different lens on the same underlying fact, and in practice the determinant and rank conditions are the most commonly used computational tests.

Why the Determinant Test Works

The determinant is a multilinear, alternating function of the columns of M that vanishes exactly when the columns are linearly dependent; linear dependence of the columns is precisely the statement that φ fails to be injective, since a nontrivial linear combination of columns equaling zero corresponds to a nonzero vector in the kernel. The determinant therefore serves as a single scalar certificate for invertibility, reducing a structural question about the whole map to the evaluation of one number.

Explicit Inverse Formulas

The Adjugate Formula

When M is invertible, its inverse has the explicit closed form

where adj(M), the adjugate, is the transpose of the matrix of cofactors of M. This formula makes explicit why nonvanishing determinant is not merely a diagnostic but the literal quantity by which the adjugate must be divided to produce the inverse, and why invertibility fails exactly at the singular locus where this division becomes impossible.

Cramer's Rule as an Invertibility Consequence

A direct consequence of invertibility is that the linear system Mx = b has the unique explicit solution given by Cramer's rule, expressing each component x_i as a ratio of determinants formed by replacing the i-th column of M with b. This is the invertibility property translated into a formula for solving tensor-index equations such as raising an index via a metric, where the "system" being solved is the defining relation between a tensor's components and the components of its image under the invertible map.

Invertibility of Maps Induced on Tensor Constructions

Invertibility Is Preserved Under Tensor Power

If φ : V → V is invertible, its induced tensor power map φ^{⊗k} : V^{⊗k} → V^{⊗k} is also invertible, with inverse (φ^{-1})^{⊗k}, since composing φ^{⊗k} with (φ^{-1})^{⊗k} on simple tensors gives φ(φ^{-1}(v_1)) ⊗ ... ⊗ φ(φ^{-1}(v_k)) = v_1 ⊗ ... ⊗ v_k, and this composition extends linearly to the identity on all of V^{⊗k}.

Invertibility of the Induced Dual Map

The pullback φ^* : V* → V* induced by an invertible φ is likewise invertible, with inverse (φ^{-1})^*, since (φ^{-1})^* ∘ φ^* = (φ ∘ φ^{-1})^* = id^* = id. This is the precise justification for why the contragredient transformation used on covariant tensor indices is always well-defined whenever the underlying change-of-basis map is invertible, which it always is by definition of a basis change.

Invertibility on Direct Sums

If φ : V → V and ψ : W → W are both invertible, the direct sum map φ ⊕ ψ : V ⊕ W → V ⊕ W is invertible, with inverse φ^{-1} ⊕ ψ^{-1}. Invertibility, unlike more delicate properties such as diagonalizability, is closed under nearly every standard construction used to build new tensor spaces from old ones, which is why it can generally be checked once on the base map and inherited automatically by every induced construction.

Invertibility of Bilinear Forms and the Metric Tensor

Nondegeneracy as Invertibility of the Associated Matrix

A symmetric bilinear form g on V, such as a metric tensor, is called nondegenerate exactly when its component matrix g_{ij} is invertible, that is, det(g_{ij}) ≠ 0. This is a direct application of matrix invertibility to a covariant (0, 2)-tensor rather than to a (1, 1) linear map, and it is the condition required before the metric can be used to identify V with V*.

Index Raising and Lowering Require Invertibility

The operation of lowering an index, v_i = g_{ij} v^j, and the operation of raising an index, v^i = g^{ij} v_j, are mutual inverses precisely because g^{ij}, the components of the inverse matrix (g_{ij})^{-1}, exist only when g_{ij} is invertible. A degenerate metric, one with vanishing determinant, cannot be used to raise indices at all, since there is no well-defined g^{ij} to apply; this is why nondegeneracy is stated as a standing hypothesis whenever index raising and lowering are introduced in tensor algebra.

Consequences of Failed Invertibility

When a linear map or bilinear form fails to be invertible, the tensor constructions that depend on invertibility, contragredient transformation, tensor power inversion, and index raising and lowering, simply do not exist for that map or form; the associated tensor object is then called singular or degenerate, and any formula that presumes an inverse must be replaced by a more delicate treatment, such as a pseudo-inverse, or restricted to the subspace on which the map or form does remain nondegenerate.