10.14.3 Tensor Inverse Jacobian Matrix Product Identity

The Tensor Inverse Jacobian Matrix Product Identity connects inverse operations with Jacobian matrices in tensor algebra, defining transformation relationships.

Tensor Inverse Jacobian Matrix Product Identity is the algebraic statement that multiplying the forward Jacobian matrix by its inverse, in either order, yields the identity matrix, expressed through the Kronecker delta, and it is the identity that certifies the inverse Jacobian is genuinely the inverse rather than merely a related matrix of partial derivatives.

Statement of the Identity

Right Product

Multiplying the forward Jacobian by the inverse Jacobian, summing over the shared middle index, produces the Kronecker delta on the remaining indices:

\sum_{k}^{n} J_{i}^{k} {(J^{- 1})}_{k}^{j} = δ_{i}^{j}

Left Product

Multiplying in the opposite order, with the inverse Jacobian first and the forward Jacobian second, produces the same Kronecker delta on its own pair of indices:

\sum_{k}^{n} {(J^{- 1})}_{i}^{k} J_{k}^{j} = δ_{i}^{j}

For square Jacobian matrices, both products give the identity simultaneously, since a one-sided matrix inverse of a square matrix is automatically a two-sided inverse.

Chain Rule Derivation

Composition of Coordinate Maps

The identity follows directly from applying the multivariable chain rule to the composition of the forward coordinate map with its own inverse map, which returns the original coordinates unchanged:

x^{i} = x^{i} ({\bar{x}}^{1} (x), \dots, {\bar{x}}^{n} (x))

Differentiating the Identity Map

Differentiating both sides of this identity with respect to $x^{j}$ and applying the chain rule to the right-hand side produces exactly the summed product of the forward and inverse Jacobian entries, which must equal the derivative of $x^{i}$ with respect to $x^{j}$ , namely the Kronecker delta, since the original coordinates are mutually independent.

Consequence for Tensor Contractions

Preserving Contracted Scalars

The product identity is precisely what guarantees that a full contraction between a covariant and a contravariant tensor component remains unchanged under a change of basis. Starting from the new-basis contraction and substituting both transformation laws:

\sum_{j}^{n} {\bar{V}}^{j} {\bar{W}}_{j} = \sum_{i}^{n} \sum_{k}^{n} \sum_{j}^{n} J_{i}^{j} {(J^{- 1})}_{j}^{k} V^{i} W_{k}

the sum over $j$ collapses through the product identity to a Kronecker delta between $i$ and $k$ , leaving exactly the original-basis contraction and confirming the invariance of the scalar.

Matrix Notation View

Compact Form

Written as ordinary matrix multiplication rather than indexed sums, the identity reads:

J J^{- 1} = J^{- 1} J = I

where $I$ denotes the identity matrix of the same size as the number of coordinates, and this compact form is the reason the inverse Jacobian is legitimately called a matrix inverse rather than an independently defined array of derivatives.

Diagram of the Cancellation

Round Trip Through Both Maps

Non-Square and Singular Cases

Requirement of a Square Non-Singular Jacobian

The product identity in its standard two-sided form requires the forward Jacobian to be a square matrix with non-zero determinant, so that the number of original coordinates equals the number of new coordinates and the coordinate map is locally invertible; when these conditions fail, no genuine two-sided inverse Jacobian exists, and the product identity does not hold in the strict Kronecker-delta form.

Local Validity

Even when the identity does hold, it holds pointwise, at each point where the Jacobian determinant is non-zero, and both the forward and inverse Jacobian entries are, in general, functions of position, so the product identity must be understood as a statement that is true at every point individually rather than as a single constant matrix equation valid everywhere.