Quadratic Forms

This chapter systematically studies the definition, matrix representation, canonical form, positive definiteness, and congruence transformation of quadratic forms.

Definition and Matrix Representation of Quadratic Forms

• Quadratic form: $Q(x) = \sum a_{ij}x_ix_j$
• Matrix representation: $Q(x) = x^T A x$

Congruence Transformation and Canonical Form

• Congruence transformation: $B = P^TAP$
• Canonical form: $Q(x) = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \dots + \lambda_n y_n^2$
• Inertia theorem

Positive Definiteness

• Positive definite quadratic form: $Q(x) > 0$ (for all nonzero $x$)
• Criterion: all leading principal minors are positive

Exercises

1. Write the matrix representation of the quadratic form $Q(x_1,x_2) = 2x_1^2 + 4x_1x_2 + 3x_2^2$.
2. Diagonalize $Q(x_1,x_2) = 2x_1^2 + 4x_1x_2 + 3x_2^2$ by an orthogonal transformation.
3. Determine whether the quadratic form $Q(x_1,x_2) = x_1^2 + 2x_2^2$ is positive definite.
4. State the inertia theorem.
5. Determine whether $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ is positive definite.