15.15.3 Tensor Symmetric Matrix Diagonalization Context

Tensor Symmetric Matrix Diagonalization Context is the account of the conditions, procedures, and interpretive framework under which a symmetric matrix, understood as a symmetric order-two tensor, can be brought to diagonal form, and of how this diagonalization is used as the working tool underlying the Matrix Case's spectral theorem, rank computations, and symmetric decomposition.

Setting Up the Diagonalization Problem

What Diagonalization Means Here

Diagonalizing a symmetric matrix T means finding an invertible matrix P such that the transformed array,

is diagonal, i.e., has zero entries off the main diagonal. This is the congruence transformation appropriate to a tensor with two covariant indices, as established under the Tensor Role of the symmetric matrix, and it should be distinguished from diagonalization by similarity, which is the transformation appropriate to a matrix representing a linear map rather than a bilinear form.

The Two Diagonalization Questions

Two related but logically distinct diagonalization statements are relevant in this context: the existence of some invertible P, not required to be orthogonal, achieving diagonal form (a purely algebraic congruence statement, valid over any field of characteristic not two), and the stronger existence of an orthogonal P achieving diagonal form with the diagonal entries being the genuine eigenvalues of T (the spectral theorem, valid over the real numbers using the standard inner product). Both statements hold for every symmetric matrix, but they diagonalize T in different senses and produce different diagonal entries in general.

The Orthogonal Diagonalization Procedure

Existence via the Spectral Theorem

Over the real numbers, every symmetric matrix T admits an orthonormal basis of eigenvectors, and expressing T in this basis yields

where Q is orthogonal, meaning its transpose equals its inverse, and Lambda is diagonal with the real eigenvalues of T along the diagonal. This is simultaneously a diagonalization by congruence, since Q-transpose equals Q-inverse, and a diagonalization by similarity, and the coincidence of the two notions in the orthogonal case is precisely why the spectral theorem is available only for symmetric matrices and not for general square matrices.

Computing the Diagonalization

Practically, the diagonalization is obtained by finding the roots of the characteristic polynomial of T to determine the eigenvalues, then solving the corresponding homogeneous linear systems to find an eigenvector for each eigenvalue, and, when an eigenvalue is repeated, choosing an orthonormal basis within its eigenspace by a Gram-Schmidt-type procedure so that the full collection of eigenvectors is mutually orthonormal.

Interpretive Role Within Symmetric Tensor Theory

Diagonalization as Symmetric Decomposition

Once diagonalized, T is exhibited as a weighted sum of pure power forms built from the orthonormal eigenvectors,

which is exactly a symmetric decomposition of T in the sense developed generally for symmetric tensors. Diagonalization in the Matrix Case is thus not a separate topic from decomposition theory but its complete and constructive solution at order two, contrasting with the more delicate Reconstruction procedures required once the order exceeds two.

Rank and the Diagonalization Context

The number of nonzero entries appearing on the diagonal after orthogonal diagonalization equals the rank of T in every sense simultaneously discussed in the Matrix Case: the linear-algebraic matrix rank, the ordinary tensor rank, and the symmetric tensor rank, since the eigendecomposition already exhibits a minimal symmetric decomposition once zero terms are discarded. This triple coincidence of rank notions is a direct, immediate consequence of the diagonalization being both orthogonal and congruent simultaneously.

Definiteness Read Off the Diagonal

The signs of the diagonal entries after orthogonal diagonalization determine the definiteness classification of T: all positive signs correspond to positive definiteness, all negative to negative definiteness, mixed signs to indefiniteness, and any zero entries to a degenerate (rank-deficient) case. This sign pattern is invariant under further orthogonal change of basis and is the content of Sylvester's law of inertia applied within the Diagonalization Context.

Limits of the Diagonalization Context

Simultaneous Diagonalization of Multiple Matrices

A single symmetric matrix is always orthogonally diagonalizable, but two symmetric matrices are simultaneously orthogonally diagonalizable by the same Q only under an additional commutativity condition. When this condition fails, generalized eigenvalue techniques, which diagonalize one matrix relative to another rather than relative to the identity, are used instead, and these generalized techniques are precisely the tool later adapted, under Reconstruction, to recover the term set of order-three symmetric tensors from an associated pencil of matrices.

Absence of a Direct Higher-Order Analogue

The Diagonalization Context is specific to order two because it relies on the spectral theorem, which has no literal extension to symmetric tensors of order three or higher: such tensors do admit notions of tensor eigenvalues and eigenvectors, but these do not, in general, yield an additive orthogonal decomposition into pure power forms with the same guarantees of existence, orthogonality, and uniqueness enjoyed in the Matrix Case.