Editing Schur complement (section)

== Conditions for positive definiteness and semi-definiteness ==

Let ''X'' be a symmetric matrix of real numbers given by
<math display="block">X = \left[\begin{matrix} A & B \\ B^\mathrm{T} & C\end{matrix}\right].</math>
Then
* If ''A'' is invertible, then ''X'' is positive definite if and only if ''A'' and its complement ''X/A'' are both positive definite:<ref name="Zhang 2005"></ref>{{rp|34}}
:<math display="block">X \succ  0 \Leftrightarrow  A \succ  0, X/A = C - B^\mathrm{T} A^{-1} B \succ  0.</math>
* If ''C'' is invertible, then ''X'' is positive definite if and only if ''C'' and its complement ''X/C'' are both positive definite:
:<math display="block">X \succ  0 \Leftrightarrow C \succ  0, X/C = A - B C^{-1} B^\mathrm{T} \succ  0.</math>
* If ''A'' is positive definite, then ''X'' is positive semi-definite if and only if the complement ''X/A'' is positive semi-definite:<ref name="Zhang 2005"/>{{rp|34}}
:<math display="block">\text{If } A \succ 0, \text{ then } X \succeq 0 \Leftrightarrow X/A = C - B^\mathrm{T} A^{-1} B \succeq 0.</math>
* If ''C'' is positive definite, then  ''X'' is positive semi-definite if and only if the complement ''X/C'' is positive semi-definite: 
:<math display="block">\text{If } C \succ 0,\text{ then } X \succeq 0 \Leftrightarrow X/C = A - B C^{-1} B^\mathrm{T} \succeq 0.</math>

The first and third statements can be derived<ref name="Boyd 2004">Boyd, S. and Vandenberghe, L. (2004), "Convex Optimization", Cambridge University Press (Appendix A.5.5)</ref> by considering the minimizer of the quantity
<math display="block">u^\mathrm{T} A u + 2 v^\mathrm{T} B^\mathrm{T} u + v^\mathrm{T} C v, \,</math>
as a function of ''v'' (for fixed ''u'').

Furthermore, since
<math display="block">
                      \left[\begin{matrix} A & B \\ B^\mathrm{T} & C \end{matrix}\right] \succ 0
  \Longleftrightarrow \left[\begin{matrix} C & B^\mathrm{T} \\ B & A \end{matrix}\right] \succ 0
</math>
and similarly for positive semi-definite matrices, the second (respectively fourth) statement is immediate from the first (resp. third) statement.

There is also a sufficient and necessary condition for the positive semi-definiteness of ''X'' in terms of a generalized Schur complement.<ref name="Zhang 2005" /> Precisely,
* <math>X \succeq 0 \Leftrightarrow A \succeq  0, C - B^\mathrm{T} A^g B \succeq 0, \left(I - A A^{g}\right)B = 0 \, </math> and
* <math>X \succeq 0 \Leftrightarrow C \succeq  0, A - B C^g B^\mathrm{T} \succeq  0, \left(I - C C^g\right)B^\mathrm{T} = 0, </math>

where <math>A^g</math> denotes a [[generalized inverse]] of <math>A</math>.