15.16.2 Tensor Quadratic Form Component Expression

The Tensor Quadratic Form Component Expression represents a structured way to express quadratic forms using tensor components in multilinear algebra.

Tensor Quadratic Form Component Expression is the explicit, coordinate-level formula that writes the quadratic form associated with a symmetric order-two tensor as a sum of terms built from the tensor's individual components and the coordinates of the vector argument, and it is the working expression through which the abstract Tensor Quadratic Form Relation is turned into a concrete polynomial that can be computed, expanded, and manipulated.

Deriving the Component Expression

Starting from the Tensor Contraction

The Tensor Quadratic Form Relation defines the quadratic form Q associated with a symmetric tensor T by the contraction Q(x) equals T(x, x). Introducing coordinates by fixing a basis of the underlying n-dimensional vector space, so that x has coordinates x_1 through x_n and T has components T_ij, the bilinear contraction expands into a double sum:

Q (x) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} T_{i j} x_{i} x_{j}

This double sum is the Component Expression in its most direct form, valid before any simplification using the symmetry of T has been applied.

Collapsing Using Symmetry

Because T_ij equals T_ji, the terms with i different from j pair up, and the double sum collapses into a form separating diagonal and off-diagonal contributions:

Q (x) = \sum_{i = 1}^{n} T_{i i} x_{i}^{2} + 2 \sum_{i < j}^{} T_{i j} x_{i} x_{j}

This is the standard, fully reduced Component Expression: each diagonal entry of T contributes a pure squared term, and each off-diagonal entry, taken once, contributes twice to a cross term because of the symmetry-induced pairing.

Small Cases Written Out

Two Variables

For n equal to two, with tensor components T_11, T_12 equal to T_21, and T_22, the Component Expression reads

Q (x) = T_{11} x_{1}^{2} + 2 T_{12} x_{1} x_{2} + T_{22} x_{2}^{2}

which is the familiar general binary quadratic form, showing directly how the three independent components of a two-by-two symmetric tensor correspond to the three coefficients of a general quadratic in two variables.

Three Variables

For n equal to three, the six independent components of the symmetric tensor produce the six-term expression

Q (x) = T_{11} x_{1}^{2} + T_{22} x_{2}^{2} + T_{33} x_{3}^{2} + 2 T_{12} x_{1} x_{2} + 2 T_{13} x_{1} x_{3} + 2 T_{23} x_{2} x_{3}

illustrating the general pattern that the number of terms equals the dimension count of symmetric order-two tensors, n times n plus one, divided by two.

Reading Off the Tensor from the Expression

Extraction of Diagonal Entries

Given a quadratic form presented purely as a polynomial expression, the diagonal entries of the associated tensor are read off directly as the coefficients of the pure squared terms x_i squared, with no adjustment needed.

Extraction of Off-Diagonal Entries

The off-diagonal entries require halving the coefficient of the corresponding cross term, since each off-diagonal entry contributes with a factor of two in the standard Component Expression: the coefficient of the cross term in x_i and x_j, for i different from j, must be divided by two to recover T_ij. This halving step is the concrete, computational counterpart of the abstract polarization identity used in the Tensor Quadratic Form Relation to recover a tensor from its associated form.

Practical Uses of the Component Expression

Verifying and Manipulating Forms

The Component Expression is the form in which quadratic forms are typically presented in applications such as optimization objectives, statistical models, and physical energy functionals, and converting such an expression into its associated symmetric tensor (matrix) is the standard first step before applying diagonalization, definiteness testing, or symmetric decomposition procedures.

Consistency Check via Substitution

Because the Component Expression is derived directly from the contraction defining the Tensor Quadratic Form Relation, substituting the coordinates of any specific vector into the expression must reproduce exactly the value obtained by contracting the tensor with that vector twice, and this substitution check is the simplest way to confirm that a claimed tensor-to-form or form-to-tensor conversion has been carried out correctly.