Linear Algebra
8.1612 min read

Definiteness

The definiteness of a symmetric matrix AA classifies the quadratic form f(x)=x⊤Axf(\mathbf{x}) = \mathbf{x}^\top A \mathbf{x} by its sign behavior across all nonzero inputs. There are five cases based on eigenvalue signs:

Positive definite (PD): all eigenvalues strictly positive. Then f(x)>0f(\mathbf{x}) > 0 for all x≠0\mathbf{x} \neq \mathbf{0} (bowl opening upward, unique minimum at 0\mathbf{0}). Positive semidefinite (PSD): all eigenvalues ≥0\geq 0. Then f(x)≥0f(\mathbf{x}) \geq 0 for all x\mathbf{x} (flat or upward). Negative definite (ND): all eigenvalues strictly negative. Negative semidefinite (NSD): all eigenvalues ≤0\leq 0. Indefinite: mixed signs (some positive, some negative eigenvalues) — saddle shape.

Definiteness is the single most important classification of a symmetric matrix for optimization: a PD matrix has a unique minimum; a PSD matrix may have infinitely many minima; an indefinite matrix has no minimum.

Formal View

Definition 8.8 — Definiteness of a Symmetric Matrix
A symmetric matrix AA is:\n- Positive definite (PD): x⊤Ax>0\mathbf{x}^\top A \mathbf{x} > 0 for all x≠0\mathbf{x} \neq \mathbf{0}   ⟺  \iff all λi>0\lambda_i > 0.\n- Positive semidefinite (PSD): x⊤Ax≥0\mathbf{x}^\top A \mathbf{x} \geq 0 for all x\mathbf{x}   ⟺  \iff all λi≥0\lambda_i \geq 0.\n- Negative definite (ND): x⊤Ax<0\mathbf{x}^\top A \mathbf{x} < 0 for all x≠0\mathbf{x} \neq \mathbf{0}   ⟺  \iff all λi<0\lambda_i < 0.\n- Negative semidefinite (NSD): x⊤Ax≤0\mathbf{x}^\top A \mathbf{x} \leq 0 for all x\mathbf{x}   ⟺  \iff all λi≤0\lambda_i \leq 0.\n- Indefinite: x⊤Ax\mathbf{x}^\top A \mathbf{x} takes both positive and negative values   ⟺  \iff some λi>0\lambda_i > 0 and some λi<0\lambda_i < 0.

Interactive Visualization

Definiteness Classifier

Why This Matters

Positive definiteness is the matrix condition equivalent to "the quadratic has a unique minimum" — essential for optimization.

  • The Hessian of a function at a critical point is PD iff the critical point is a strict local minimum.
  • Kernel matrices in machine learning are always PSD by construction.
  • Control theory: a system is stable iff a certain matrix (Lyapunov function matrix) is PD.

Quiz

Question 1

A symmetric matrix with eigenvalues {4,0.5,0.1}\{4, 0.5, 0.1\} is:

Question 2

Every positive definite matrix is also positive semidefinite.

Common Mistakes

  • Confusing PD (>0>0 strictly) with PSD (≥0\geq 0) — a matrix with a zero eigenvalue is PSD but NOT PD.
  • Checking only diagonal entries of AA to determine definiteness — you must check eigenvalues (or use the principal minors test) for the full picture.