15.2.2 Tensor Symmetric Form Area

Tensor Symmetric Form Area is the branch of application in which symmetric tensors of degree two serve as the algebraic model for symmetric bilinear forms and their associated quadratic forms, providing the setting for diagonalization, classification by signature, and congruence transformations that underlie much of classical linear algebra.

The Matrix Model of a Symmetric Form

Symmetric Matrices as Degree-Two Symmetric Tensors

Relative to a basis of $V$, a symmetric bilinear form $B$ is represented by a matrix $A$ satisfying $A^T=A$, with

and this symmetry of the matrix is exactly the component-level manifestation, discussed at the level of the symmetric component scope, of $B$ being an element of ${\text{Sym}}^2\left(V^{\ast }\right)$.

Recovering the Form From the Quadratic Function

Whenever the characteristic of the field is not two, the polarization identity

recovers $B$ entirely from its associated quadratic form $Q\left(v\right)=B(v,v)$, confirming that the symmetric tensor $B$ and the quadratic function $Q$ carry exactly the same information in this setting.

Diagonalization

Congruence Transformations

Two symmetric matrices $A$ and $\tilde{A}$ represent the same bilinear form in different bases exactly when they are related by congruence,

for an invertible change of basis matrix $P$, distinct from the ordinary similarity transformation used for the matrix of a linear operator, since $B$ transforms with two copies of the same change of basis matrix rather than one copy and its inverse.

Diagonalizability by Congruence

Every symmetric matrix over a field of characteristic not two is congruent to a diagonal matrix, a fact proven directly using the symmetric structure, typically by an inductive argument completing the square one variable at a time; this guarantees that every symmetric bilinear form can be brought, by a suitable choice of basis, into the simple diagonal form $Q\left(v\right)=\sum _i{\lambda }_i{v}_i^2$.

Classification by Signature

Sylvester's Law of Inertia

Over the real numbers, the number of positive, negative, and zero diagonal entries obtained by diagonalizing a symmetric form is an invariant of the form itself, independent of which specific diagonalizing basis is chosen; this triple of counts, called the signature, is the complete invariant classifying real symmetric bilinear forms up to congruence.

Positive Definiteness

A symmetric form with signature consisting entirely of positive entries is called positive definite, and such forms are exactly the ones arising as inner products, connecting the classification of symmetric tensors of degree two directly to the theory of inner product spaces and, through the resulting positive-definite covariance matrices, to statistics.

The Spectral Perspective

Orthogonal Diagonalization

When the underlying field is the real numbers and a fixed inner product is also available, a symmetric matrix can be diagonalized not merely by an arbitrary congruence but by an orthogonal congruence, in which case the congruence and similarity transformations coincide; this is the spectral theorem for symmetric matrices, and it identifies the diagonal entries obtained this way with the actual eigenvalues of the matrix, a much stronger statement than the mere existence of some diagonalizing congruence.

Consequence for the Signature

Since orthogonal congruence is a similarity transformation, the eigenvalues themselves, not merely their signs, are basis-independent invariants of the symmetric form in this setting, and the signature can be read off directly as the count of positive, negative, and zero eigenvalues.

Rank as an Invariant

Rank of the Form

The rank of the matrix $A$ representing a symmetric bilinear form is likewise a congruence invariant, equal to the number of nonzero entries in any diagonalization, and together with the signature over the reals, or with the rank alone over an arbitrary field of characteristic not two, this data completely determines the equivalence class of the symmetric tensor under change of basis.