15.15.4 Tensor Symmetric Matrix Quadratic Form Relation

Symmetric matrices and tensors link quadratic forms through bilinear transformations in algebraic structures.

Tensor Symmetric Matrix Quadratic Form Relation is the correspondence identifying a symmetric order-two tensor with the homogeneous degree-two polynomial function it produces when evaluated on a single repeated vector argument, together with the precise rules by which the quadratic form determines the matrix and the matrix determines the form.

From Matrix to Quadratic Form

The Evaluation Map

Given a symmetric matrix T over an n-dimensional space, and a vector x with coordinates x_1 through x_n, the associated quadratic form is obtained by evaluating the tensor twice on x:

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

which is exactly the pure power form of x under the tensor T, in the sense of the general symmetric decomposition theory, restricted to order two. Every entry of T off the main diagonal contributes twice to Q, once through each ordering of the index pair, which is the direct manifestation of the Component Constraint at the level of the resulting polynomial.

Homogeneity and Degree

The function Q is homogeneous of degree two, meaning Q(cx) equals c squared times Q(x) for every scalar c, and it contains no terms of degree different from two; this is the order-two instance of the general correspondence between symmetric tensors of order d and homogeneous polynomials of degree d used throughout Tensor Symmetric Decomposition Structure.

From Quadratic Form to Matrix

The Polarization Identity

Conversely, any homogeneous degree-two polynomial Q determines a unique symmetric matrix T from which it arises, recovered through the polarization identity,

T_{i j} = \frac{1}{2} \frac{\partial^{2} Q}{\partial x_{i} \partial x_{j}}

valid in any characteristic different from two, which extracts the mixed second partial derivatives of Q and assembles them, with the appropriate normalizing factor, into a matrix that is automatically symmetric because mixed partial derivatives commute.

The Associated Hessian Matrix

The matrix obtained this way is, up to the constant factor of one half depending on convention, exactly the Hessian matrix of Q at any point, since Q is homogeneous of degree two and its second derivative is therefore constant. This identifies the Quadratic Form Relation as the source of the Hessian matrices used throughout multivariable calculus and optimization to test the nature of critical points.

Shared Invariants

Rank as a Common Invariant

The rank of T, understood equivalently as the linear-algebraic matrix rank, the ordinary tensor rank, and the symmetric tensor rank (all coinciding in the Matrix Case), equals the number of variables that genuinely appear in Q after an invertible linear change of coordinates has been used to bring Q to a diagonal sum of squares. Coordinates in which Q fails to depend at all correspond exactly to vectors in the radical of T.

Definiteness and the Shape of Level Sets

The sign pattern of the eigenvalues of T, obtained via the Diagonalization Context, determines whether Q is positive definite, negative definite, or indefinite, and this in turn determines the geometric shape of the level sets of Q: positive definiteness produces ellipsoidal level sets, indefiniteness produces hyperboloidal ones, and the presence of zero eigenvalues produces degenerate, cylinder-like level sets extending along the radical directions.

Applications of the Relation

Second-Order Optimality Conditions

At a critical point of a smooth function, the Quadratic Form Relation applied to the Hessian matrix of the function determines the local behavior of the function through the definiteness of the associated quadratic form: positive definiteness certifies a local minimum, negative definiteness a local maximum, and indefiniteness a saddle point, making this relation the direct algebraic tool behind the second-derivative test.

Conics, Quadrics, and Classification

Plane conics and higher-dimensional quadric hypersurfaces are level sets of quadratic forms (possibly with added linear and constant terms), and their classification into ellipses, parabolas, hyperbolas, and their higher-dimensional analogues proceeds by diagonalizing the symmetric matrix associated to the quadratic part via the Quadratic Form Relation, reducing the geometric classification problem entirely to the eigenvalue and rank data of the corresponding matrix.

Statistical Interpretation

When T is a covariance matrix, the associated quadratic form gives the squared Mahalanobis-type distance used to measure statistical deviation relative to the covariance structure, and the ellipsoidal level sets of positive definite covariance matrices are precisely the confidence ellipsoids used in multivariate statistical inference.