16.14.1 Tensor Determinant Multilinear Form

The Tensor Determinant Multilinear Form generalizes determinant properties to tensor spaces, capturing multilinear invariants across multi-dimensional arrays.

Tensor Determinant Multilinear Form is the property of the determinant that treats it as a function of n vector arguments, each argument corresponding to one row or column of a square matrix, such that the function is linear in every argument when all the other arguments are held fixed. It is the foundational structural property that, combined with alternation and normalization, uniquely characterizes the determinant among all possible functions of n vectors.

Statement of Multilinearity

Linearity in a Single Argument

Viewing the determinant as a function of its column vectors v₁, v₂, ..., vₙ, multilinearity means that fixing all but one column and varying that column linearly causes the determinant to respond linearly. For the i-th argument:

det (v_{1}, \dots, a u_{i} + b w_{i}, \dots, v_{n}) = a \cdot det (v_{1}, \dots, u_{i}, \dots, v_{n}) + b \cdot det (v_{1}, \dots, w_{i}, \dots, v_{n})

This must hold simultaneously and independently for every one of the n argument positions, which is what distinguishes multilinearity from ordinary linearity of a single-variable function.

Distinguishing From Total Linearity

It is important that multilinearity concerns linearity in each argument separately, not linearity of the determinant as a function of the entire matrix considered as a single vector. The determinant is not linear over the whole matrix at once; scaling every entry of an n by n matrix by a constant c scales the determinant by cⁿ, not by c, which reflects the fact that all n arguments are being scaled simultaneously.

Consequences of Multilinearity

Scalar Factor Extraction

A direct consequence of linearity in a single argument is that a scalar multiplying any one column can be factored out of the determinant entirely:

det (v_{1}, \dots, c v_{i}, \dots, v_{n}) = c \cdot det (v_{1}, \dots, v_{i}, \dots, v_{n})

This underlies the familiar row and column scaling rules used when simplifying determinants by hand.

Additivity Across a Column Sum

If one column is expressed as a sum of two vectors, the determinant splits into a sum of two determinants, each sharing all other columns unchanged. This additive splitting is the mechanism behind cofactor expansion, where a matrix's determinant is broken down by expanding along a row or column into a sum of smaller determinants weighted by matrix entries.

Expansion Along a Row or Column

Repeatedly applying multilinearity to each standard basis component of a chosen row or column produces the cofactor expansion formula:

det (A) = \sum_{j = 1}^{n} (- 1)^{i + j} a_{i j} M_{i j}

where Mᵢⱼ denotes the minor obtained by deleting row i and column j. This expansion is a direct computational consequence of the multilinear structure applied one argument at a time.

Multilinearity Combined With Alternation

The Full Characterization

Multilinearity alone does not uniquely determine the determinant; it must be paired with the alternating property, which states that the determinant vanishes whenever two columns are equal, and with a normalization condition fixing the value on the identity matrix. Together these three properties uniquely single out the determinant among all functions of n vectors on an n-dimensional space.

Role Within the Exterior Algebra

Multilinearity is exactly the property shared by the tensor product before antisymmetrization is imposed. The determinant can be understood as the unique alternating multilinear form obtained by projecting the fully multilinear tensor structure onto its totally antisymmetric component, which is precisely the top exterior power of the space, linking the multilinear form directly to the wedge product construction.

Significance of Multilinearity

Multilinearity establishes the determinant as a function that behaves predictably and computably with respect to elementary operations on the vectors or rows that define it. It underlies cofactor expansion, scalar factoring rules, and additive column splitting, and together with alternation and normalization it provides the complete axiomatic characterization from which every classical determinant identity can be derived.