Positive Definite Matrix¶

Formula¶

\[ A \succ 0 \iff x^\top A x > 0 \quad \forall x\ne 0 \]

Parameters¶

\(A\): symmetric matrix (in the usual real-valued definition)
\(x\): nonzero vector

What it means¶

A positive definite matrix defines a strictly positive quadratic form.

What it's used for¶

Convex quadratic optimization.
Covariance/kernel matrices and Cholesky factorization.

Key properties¶

Symmetric positive definite matrices have positive eigenvalues.
Admit Cholesky factorization \(A=LL^\top\).

Common gotchas¶

Positive entries do not imply positive definiteness.
Must distinguish positive semidefinite vs definite.

Example¶

The identity matrix \(I\) is positive definite because \(x^\top I x=\|x\|^2>0\).

How to Compute (Pseudocode)¶

Input: symmetric matrix A
Output: positive-definite check (true/false)

choose a test method
  examples: try Cholesky factorization, or check all eigenvalues > 0
if the test succeeds:
  return true
else:
  return false

Complexity¶

Time: Depends on the test method (dense Cholesky or eigendecomposition methods are typically cubic in matrix size)
Space: Depends on matrix storage and factorization/eigensolver workspaces
Assumptions: Symmetric real-matrix setting; floating-point tests use tolerances for near-zero eigenvalues/pivots