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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

1.1

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 $λ_{1}, \dots, λ_{n}$ .

L1L2

2.1

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.

L3step 1.1

3.1

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

step 2.1∎

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)