Quadratic Form
1. Definition
A quadratic form in n variables is a homogeneous polynomial of degree 2 — every term is of second degree. In matrix form it is written with a symmetric matrix:
For two variables the general form is Q = ax² + 2hxy + by² with matrix
and for three variables Q = a₁₁x² + a₂₂y² + a₃₃z² + 2a₁₂xy + 2a₂₃yz + 2a₃₁zx with the symmetric matrix A = [a₁₁ a₁₂ a₃₁ ; a₁₂ a₂₂ a₂₃ ; a₃₁ a₂₃ a₃₃].
2. How to Write the Matrix of a Quadratic Form
- The coefficient of each square (x², y², z²) goes on the diagonal.
- A cross term 2axy is split equally: a goes in position (1,2) and a in position (2,1). This equal split is exactly what makes A symmetric — and it is the point of the whole construction.
Q = 2x² + 3y² + 5z² + 4xy + 6yz - 8zx
[ 2 2 -4 ] (4xy → 2,2 : 6yz → 3,3 : -8zx → -4,-4)
A = [ 2 3 3 ]
[ -4 3 5 ]
Check: XᵀAX = 2x² + 3y² + 5z² + 2(2xy) + 2(3yz) + 2(-4zx) ✓
3. Nature (Classification) of a Quadratic Form
The nature is decided by the signs of the eigenvalues of A:
| Nature | Eigenvalues | Meaning |
|---|---|---|
| Positive definite | all λ > 0 | Q > 0 for every non-zero X (a bowl touching the origin) |
| Positive semi-definite | all λ ≥ 0, at least one λ = 0 | Q ≥ 0, and Q = 0 for some non-zero X |
| Negative definite | all λ < 0 | Q < 0 for every non-zero X |
| Negative semi-definite | all λ ≤ 0, at least one λ = 0 | Q ≤ 0 |
| Indefinite | λ's of both signs | Q takes positive and negative values |
Sylvester's criterion gives the same answer without computing eigenvalues, using the leading principal minors D₁ = a₁₁, D₂ = det of the first 2 × 2 corner, D₃ = det A, …:
- Positive definite ⇔ D₁ > 0, D₂ > 0, D₃ > 0 (all positive);
- Negative definite ⇔ D₁ < 0, D₂ > 0, D₃ < 0 (signs alternate, starting negative);
- If all Dᵢ ≠ 0 but the signs follow neither pattern → indefinite;
- If some Dᵢ = 0, the test is inconclusive for definiteness (the form is then semi-definite or indefinite — decide from the eigenvalues).
4. Rank, Index and Signature
Reduce Q by a non-singular transformation X = PY to the canonical (diagonal) form
| Quantity | Definition | How to get it |
|---|---|---|
| Rank r | number of non-zero terms in the canonical form | = number of non-zero eigenvalues = ρ(A) |
| Index p | number of positive terms | = number of positive eigenvalues |
| Signature s | (number of positive terms) − (number of negative terms) | s = 2p − r |
Rank, index and signature are invariants — they do not change under any non-singular (or orthogonal) change of variables, which is why they characterise the form.
Q = 2x² + 2y² + 2z² + 2xy + 2yz + 2zx
[ 2 1 1 ]
A = [ 1 2 1 ] eigenvalues 4, 1, 1 (all positive)
[ 1 1 2 ]
⇒ NATURE : POSITIVE DEFINITE
⇒ canonical form (using the orthogonal transformation from the Diagonalization topic,
Q = [1/√3, 1/√3, 1/√3 ; 1/√2, -1/√2, 0 ; 1/√6, 1/√6, -2/√6] ) is
4Y₁² + Y₂² + Y₃²
⇒ rank r = 3 , index p = 3 , signature s = 2(3) - 3 = +3
Sylvester check: D₁ = 2 > 0 , D₂ = |2 1 ; 1 2| = 3 > 0 , D₃ = det A = 4 > 0 ✓
7. Why Quadratic Forms Matter in Engineering
Energy expressions (kinetic energy = ½XᵀMX, potential energy of a spring system), stability of equilibrium in mechanics and control theory (a positive definite energy function means a stable equilibrium — the Lyapunov method), the geometry of conics and quadrics (positive definite ↔ ellipse/ellipsoid, indefinite ↔ hyperbola/hyperboloid), principal moments of inertia, and the statistical notion of covariance being positive semi-definite.
8. Exam Points
- Define a quadratic form and write the matrix of a given form (symmetric A) — 5 marks, almost certain.
- Find the nature of a quadratic form using eigenvalues, and using Sylvester's criterion — 5-7 marks, very frequent.
- Define positive definite, semi-definite, negative definite and indefinite forms with the eigenvalue condition — 5 marks.
- Define and find rank, index and signature of a quadratic form — 5 marks, frequent.
- Classify a 3-variable form and state its rank/index/signature — 5-7 marks.
- Show that the nature of a quadratic form is unchanged under a non-singular transformation — 3-5 marks.
- Give an engineering application of quadratic forms (energy/stability) — 3 marks.