15.16.3 Tensor Quadratic Form Matrix Representation

Tensor Quadratic Form Matrix Representation is the packaging of a quadratic form's associated symmetric tensor as an explicit matrix acting on coordinate column vectors, together with the vector-matrix-vector notation this packaging makes available and the precise sense in which the symmetric matrix is singled out among all matrices that could represent the same form.

The Representation in Matrix-Vector Notation

From Contraction to Matrix Multiplication

Once a basis is fixed and the vector x is written as a column of coordinates, the contraction defining the associated quadratic form, Q(x) equals T(x, x), becomes the familiar matrix expression

where T is now the square matrix of components and the multiplication is ordinary matrix multiplication, first applying T to the column x, then pairing the result against x again via the row vector x-transpose. This is the Matrix Representation: the same object as the abstract tensor, but packaged so that ordinary matrix arithmetic computes the quadratic form directly.

Recovering Components from the Representation

Expanding the matrix product recovers exactly the Component Expression obtained earlier by direct contraction, confirming that the Matrix Representation is not an approximation or reformulation but a literal restatement of the same bilinear contraction in the language of matrix algebra, chosen because matrix algebra carries with it an extensive, independently developed computational toolkit.

Non-Uniqueness of a Representing Matrix, and the Symmetric Choice

Any Matrix, Not Just a Symmetric One, Can Represent Q

For a given quadratic form Q, the equation Q(x) equals x-transpose M x is satisfied not only by the Symmetric Bilinear Source of Q but by infinitely many matrices M, since adding to M any matrix A satisfying x-transpose A x equal to zero for all x (that is, any antisymmetric matrix, in characteristic different from two) leaves the quadratic form unchanged. This means the Matrix Representation is not unique if arbitrary matrices are allowed.

Symmetrization Selects a Canonical Representative

Among all matrices M representing a given Q, exactly one is symmetric, obtained from any representative by symmetrization:

and this symmetric representative is precisely the tensor identified as the Symmetric Bilinear Source under the Tensor Quadratic Form Relation. The Matrix Representation, as the term is used in the theory of symmetric tensors, always refers to this canonical symmetric choice, since it is the unique representative compatible with the Component Constraint and with the congruence transformation law appropriate to a genuine order-two tensor.

Operations Performed Through the Representation

Change of Basis

Under a coordinate change given by an invertible matrix P, so that x equals P times a new coordinate vector x-prime, the quadratic form transforms as

exhibiting directly, through matrix multiplication, the congruence transformation rule of the underlying symmetric tensor, and confirming that the Matrix Representation transforms exactly as required for consistency with the Tensor Role of a symmetric matrix.

Diagonalization Through the Representation

Because the Matrix Representation is an ordinary matrix, the full apparatus of the Diagonalization Context becomes directly applicable: an orthogonal P can be found so that P-transpose T P is diagonal, turning Q into a sum of squares with coefficients equal to the eigenvalues of T, and thereby exhibiting a symmetric decomposition of the quadratic form using only standard numerical linear algebra applied to the Matrix Representation.

Computational Significance

Matrix Representation as the Interface to Numerical Methods

Presenting a quadratic form through its Matrix Representation is what allows definiteness testing, eigenvalue computation, rank determination, and optimization second-order analysis to be carried out using standard, highly optimized numerical linear algebra routines, rather than requiring bespoke tensor-contraction code; the Matrix Representation is, in this sense, the practical computational interface through which the abstract Tensor Quadratic Form Relation is accessed in applied settings.

Consistency with the Broader Symmetric Tensor Framework

Every property established at the general, coordinate-free level of the Tensor Quadratic Form Relation, and every specialization discussed under the Matrix Case, is visible and verifiable directly on the Matrix Representation, making it the concrete object through which the abstract theory of symmetric tensors of order two is put into practice.