symmetric block matrix

Given a symmetric matrix block matrix of the form

where are sub-matrices of . The Schur complement of is defined as

Similarly, the Schur complement of is

If is invertible, the following properties hold:

• If , then

In words, this means that

• is positive definite if and only if
  • the diagonal block is positive definite AND
  • the corresponding Schur complement of , , is positive definite
• When is positive definite, then is positive semidefinite if and only if the corresponding Schur complement of , , is positive semi-definite
• Alternatively, change to and properties remain.

First encountered in CS164 Assignment 1