Diagonalform

Diagonalform is a term used in linear algebra to denote a diagonal representation of a quadratic form attained by a linear change of coordinates. For a real quadratic form Q(x) = x^T A x, where A is a real matrix, a diagonal form is an expression Q(y) = sum_i λ_i y_i^2 after a nonsingular change of variables y = P^{-1} x, with P^T A P = diag(λ_1, ..., λ_n).

If A is real and symmetric, the spectral theorem guarantees an orthogonal matrix P such that P^T A P is diagonal with the eigenvalues of A on the diagonal. This yields a diagonal form in a rotated coordinate system, often called principal axes.

Beyond orthogonal diagonalization, diagonal forms can be obtained by congruence for symmetric matrices. The resulting diagonal entries reflect the inertia of the form—the numbers of positive, negative, and zero directions—an invariance known as Sylvester's law of inertia.

Diagonalform is used to simplify optimization problems, solve systems of equations, and classify quadratic forms up to change of variables. It also serves as a tool in diagonalizing bilinear or quadratic forms in algebraic geometry and number theory, and provides a canonical representative of a quadratic form under suitable equivalence relations.

Related concepts include diagonalization, congruence, and the spectral theorem.