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A stronger Sylvester's criterion for positive semidefinite matrices

Mingrui Zhang, Peng Ding

TL;DR

This work strengthens Sylvester’s criterion for PSD matrices by proving that PSD can be certified by checking only $m(m+1)/2$ determinants, using consecutive principal submatrices and a new inner-saturated submatrix concept. It provides an elementwise framework that yields both PD and PSD criteria and enables a constructive path to determine feasible ranges for matrix entries. The authors extend the theory to practical tasks including PD/PSD matrix completion and nonlinear semidefinite programming, illustrating procedures and edge cases with $m=4$ and outlining a general approach for larger $m$. The results offer a scalable alternative to eigenvalue computations, with direct implications for statistics and optimization problems involving PSD constraints.

Abstract

Sylvester's criterion characterizes positive definite (PD) and positive semidefinite (PSD) matrices without the need of eigendecomposition. It states that a symmetric matrix is PD if and only if all of its leading principal minors are positive, and a symmetric matrix is PSD if and only if all of its principal minors are nonnegative. For an $m\times m$ symmetric matrix, Sylvester's criterion requires computing $m$ and $2^m-1$ determinants to verify it is PD and PSD, respectively. Therefore, it is less useful for PSD matrices due to the exponential growth in the number of principal submatrices as the matrix dimension increases. We provide a stronger Sylvester's criterion for PSD matrices which only requires to verify the nonnegativity of $m(m+1)/2$ determinants. Based on the new criterion, we provide a method to derive elementwise criteria for PD and PSD matrices. We illustrate the applications of our results in PD or PSD matrix completion and highlight their statistics applications via nonlinear semidefinite program.

A stronger Sylvester's criterion for positive semidefinite matrices

TL;DR

This work strengthens Sylvester’s criterion for PSD matrices by proving that PSD can be certified by checking only determinants, using consecutive principal submatrices and a new inner-saturated submatrix concept. It provides an elementwise framework that yields both PD and PSD criteria and enables a constructive path to determine feasible ranges for matrix entries. The authors extend the theory to practical tasks including PD/PSD matrix completion and nonlinear semidefinite programming, illustrating procedures and edge cases with and outlining a general approach for larger . The results offer a scalable alternative to eigenvalue computations, with direct implications for statistics and optimization problems involving PSD constraints.

Abstract

Sylvester's criterion characterizes positive definite (PD) and positive semidefinite (PSD) matrices without the need of eigendecomposition. It states that a symmetric matrix is PD if and only if all of its leading principal minors are positive, and a symmetric matrix is PSD if and only if all of its principal minors are nonnegative. For an symmetric matrix, Sylvester's criterion requires computing and determinants to verify it is PD and PSD, respectively. Therefore, it is less useful for PSD matrices due to the exponential growth in the number of principal submatrices as the matrix dimension increases. We provide a stronger Sylvester's criterion for PSD matrices which only requires to verify the nonnegativity of determinants. Based on the new criterion, we provide a method to derive elementwise criteria for PD and PSD matrices. We illustrate the applications of our results in PD or PSD matrix completion and highlight their statistics applications via nonlinear semidefinite program.
Paper Structure (18 sections, 7 theorems, 39 equations, 2 figures)

This paper contains 18 sections, 7 theorems, 39 equations, 2 figures.

Table of Contents

  1. Classic Sylvester's criterion and our new result
  2. Review of Sylvester's criterion
  3. Statement of the new result
  4. Toward elementwise characterization of PD and PSD matrices
  5. Theoretical results
  6. General method
  7. Example: $4\times 4$ matrix
  8. Application 1: PD and PSD matrix completion
  9. A general procedure for PD matrix completion
  10. A general procedure for PSD matrix completion
  11. Application 2: nonlinear semidefinite program
  12. Proofs
  13. Lemmas
  14. We first consider the case when $A$ is invertible.
  15. We then consider the case when $A$ is not invertible.
  16. ...and 3 more sections

Key Result

Theorem 1

(i) A symmetric matrix is PD if and only if all of its leading principal minors are positive. (ii) A symmetric matrix is PSD if and only if all of its principal minors are nonnegative.

Figures (2)

  • Figure 1: Partial ordering of upper triangular elements in a symmetric $4\times 4$ matrix $X$.
  • Figure 2: Feasible region of $(x_1,x_2)$ as the intersection of the inner parts of two ellipses to ensure $X$ has a PD completion.

Theorems & Definitions (20)

  • Definition 1
  • Theorem 1: Sylvester's criterion
  • Definition 2
  • Definition 3
  • Example 1
  • Example 2
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Example 3
  • ...and 10 more