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A symmetric second-order principal part has a coordinate-invariant signature normal form

Statement

Fix a point $x_{0}$ of a scalar second-order operator whose principal part is

$\sum_{i, j = 1}^{n} a_{i j} (x_{0}) \partial_{x_{i} x_{j}}, a_{i j} (x_{0}) = a_{j i} (x_{0}) .$

Then there is a linear change of coordinates near $x_{0}$ in which the frozen principal quadratic form becomes

$ξ_{1}^{2} + \dots + ξ_{p}^{2} - ξ_{p + 1}^{2} - \dots - ξ_{p + q}^{2},$

with the remaining $r = n - p - q$ directions absent. The triple $(p, q, r)$ depends only on the quadratic form, not on the chosen coordinates.

Facts & Assumptions

Given: The symmetric coefficient matrix $A = (a_{i j} (x_{0}))_{1 \leq i, j \leq n}$ of the frozen principal part at $x_{0}$ .

[L1]

The order- $2$ principal symbol is the quadratic polynomial associated to the symmetric coefficient matrix (Elliptic, hyperbolic, and parabolic principal symbols).

[L2]

A self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[L3]

A real symmetric bilinear form is congruent to exactly one diagonal form $diag (I_{p}, - I_{q}, 0_{r})$ (Sylvester's law of inertia: every real symmetric form is congruent to $diag (I_{p}, - I_{q}, 0_{r})$ , and $(p, q, r)$ is unique).

Proof

technique · direct

1.1L1L2

By [L1], the frozen principal symbol is $p_{2} (ξ) = ξ^{T} A ξ$ , and because $A$ is symmetric, [L2] gives an orthonormal eigenbasis in which $A$ is diagonal with real eigenvalues $λ_{1}, \dots, λ_{n}$ .

2.1L3step 1.1

Rescaling each coordinate with $λ_{i} \neq 0$ by $∣ λ_{i} ∣^{1 / 2}$ changes the nonzero diagonal entries to $1$ or $- 1$ and leaves the zero eigenvalues unchanged, so the principal form becomes $diag (I_{p}, - I_{q}, 0_{r})$ ; by [L3], the numbers of positive, negative, and zero directions are intrinsic.

3.1step 2.1∎

Therefore the frozen symmetric principal part has the stated signature normal form, and its signature triple is invariant under further invertible linear coordinate changes.

Depends on

Used by

Constant-coefficient second-order equations in two variables have canonical principal forms Theorem

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Victor Ivrii, Partial Differential Equations (standard reference, not scraped)