15.5.3 Tensor Symmetric Bilinear Matrix Representation

Tensor Symmetric Bilinear Matrix Representation is the encoding of a symmetric rank-2 tensor and its associated bilinear form as a square matrix that equals its own transpose, once a basis for the underlying vector space has been fixed. Choosing a basis e_1 through e_d converts the abstract tensor into a concrete array of numbers T_{ij}, arranged into rows and columns to form a matrix, and the symmetric component equality constraint T_{ij} = T_{ji} translates directly into the matrix condition that the entry in row i and column j equals the entry in row j and column i for every i and j.

This representation allows the tools of matrix algebra, such as eigenvalue decomposition, determinant computation, and rank determination, to be applied directly to the study of a symmetric bilinear form, provided the dependence of the matrix on the chosen basis is tracked correctly. The matrix representation is not intrinsic to the tensor itself but depends on the basis; changing the basis changes the matrix entries even though the underlying bilinear form, and the tensor it comes from, remain the same object.

Construction of the Matrix

Entries From Basis Vector Evaluation

Given a symmetric bilinear form B and a basis e_1 through e_d, the matrix representation M is built by evaluating B on every pair of basis vectors:

so that M is exactly the coordinate array of the tensor T in the chosen basis.

Symmetric Matrix Condition

The symmetry of B ensures M_{ij} = B(e_i, e_j) equals B(e_j, e_i) = M_{ji}, so M satisfies M = M^T, where M^T denotes the transpose obtained by reflecting M across its main diagonal. A matrix equal to its own transpose is called a symmetric matrix, and this is exactly the matrix-level statement of the tensor's equality constraint.

Evaluating the Bilinear Form Through the Matrix

Quadratic Form as a Matrix Product

For column vectors u and v representing the coordinates of two vectors in the chosen basis, the bilinear form value is computed by the matrix product:

where u^T denotes the row vector obtained by transposing u, and the right-hand side is an ordinary matrix multiplication yielding a single scalar.

Recovering Symmetry From the Product Form

Because M equals M^T, the scalar u^T M v equals its own transpose, which for a one-by-one matrix is just the scalar itself, but expanding (u^T M v)^T as v^T M^T u shows directly that u^T M v equals v^T M u whenever M is symmetric, reproducing the order invariance of the bilinear argument pair at the level of matrix computation.

Behavior Under Change of Basis

Transformation Rule

If the basis is changed by an invertible matrix A, so that new coordinates relate to old coordinates through a linear substitution, the matrix representation transforms as:

This transformation rule is called congruence, distinguishing it from the similarity transformation A^-1 M A used for linear operators, and reflects the fact that a bilinear form transforms differently under a basis change than a linear map does.

Preservation of Symmetry Under Congruence

Taking the transpose of A^T M A gives A^T M^T A, which equals A^T M A whenever M is symmetric, confirming that congruence transformations always map symmetric matrices to symmetric matrices, so the matrix representation of a symmetric bilinear form remains symmetric in every basis, even though the specific entries change.

Structural Information Encoded in the Matrix

Rank and Degeneracy

The rank of the matrix M, defined as the number of linearly independent rows or columns, is a basis-independent property of the bilinear form and indicates the dimension of the subspace on which the form acts non-trivially; a bilinear form is called degenerate when this rank is less than the dimension of the full vector space, meaning some nonzero vector v satisfies B(v, w) = 0 for every w.

Signature Under Real Diagonalization

When the matrix representation is taken over the real numbers, it can always be diagonalized by a congruence transformation into a matrix with only entries of value one, negative one, and zero along the diagonal, and the counts of positive, negative, and zero diagonal entries form the signature of the symmetric bilinear form, a basis-independent invariant guaranteed by Sylvester's law of inertia.