SUMMARY
A 2x2 matrix [x y; y z] is in the positive semidefinite (PSD) cone if and only if the conditions x >= 0, z >= 0, and xz >= y^2 are satisfied. This conclusion is derived from the properties of principal minors, which state that for a matrix to be PSD, all principal minors must be nonnegative. Specifically, for a 2x2 matrix, the relevant principal minors include the diagonal elements and the determinant, leading to the inequalities x, z >= 0 and xz - y^2 >= 0.
PREREQUISITES
- Understanding of positive semidefinite matrices
- Knowledge of principal minors in matrix theory
- Familiarity with matrix determinants
- Basic linear algebra concepts
NEXT STEPS
- Study the properties of positive semidefinite matrices in-depth
- Learn about principal minors and their significance in matrix analysis
- Explore the implications of the determinant in matrix positivity
- Investigate alternative characterizations of PSD matrices
USEFUL FOR
Mathematicians, students of linear algebra, and professionals working with matrix theory and optimization will benefit from this discussion.