This section discusses the class of positive definite matrices, which are symmetric and have the property that for all non-zero vectors \( x \), \( x^T A x > 0 \). It also defines partial positive definite matrices, where every fully specified principal submatrix must be positive definite. The key result is that a pattern \( Q \) has a positive definite completion if and only if the principal subpattern determined by its diagonal positions has a positive definite completion. Additionally, a pattern with all diagonal positions has a positive definite completion if and only if its pattern graph is chordal. The section includes examples of patterns that do and do not have positive definite completions, and references to a relevant paper by Grone, Johnson, Sá, and Wolkowicz.This section discusses the class of positive definite matrices, which are symmetric and have the property that for all non-zero vectors \( x \), \( x^T A x > 0 \). It also defines partial positive definite matrices, where every fully specified principal submatrix must be positive definite. The key result is that a pattern \( Q \) has a positive definite completion if and only if the principal subpattern determined by its diagonal positions has a positive definite completion. Additionally, a pattern with all diagonal positions has a positive definite completion if and only if its pattern graph is chordal. The section includes examples of patterns that do and do not have positive definite completions, and references to a relevant paper by Grone, Johnson, Sá, and Wolkowicz.