The following question was proposed in the NUS Q.E. in 2009: Given matrices , and . Suppose and are symmetric. Consider the following matrices
Show that is positive definite if and only if and are positive definite.
In the literature, the matric is called the Schur complement, usually, it is denoted by with respect to . In other word, is of the form . It is worth noting that the letter used in the above notation indicates the full matrix , roughly speaking, by we mean the Schur complement of with respect to .
Throughout this entry, by (resp. ) we mean that is positive definite (resp. positive semi-definite). In order to solve the above problem, one needs the following matrix identity, the Aitken block-diagonalization formula,
Now we assume and . Then the following property
holds true. Indeed, since
then for every
Note that at least or is not a zero vector so that
which proves the positive definite property of
Now by means of the above matrix identity we claims that .
Conversely, for every , one has
whenever . Thus, this and the fact that is symmetric implies that . As a consequence, exists which helps us to say that is well-defined.
Now with the help of the matrix identity, one gets
Note that, the left left side of the above identity is nothing but
which is positive by the assumption provided . Hence, if is arbitrary, the right hand side equals to
which proves that . The proof is complete.
If I have time, I will provide another proof using the Sylvester’s Law of Inertia. For your convenience regarding to the Schur complement, I prefer you to the book entitled THE SCHUR COMPLEMENT AND ITS APPLICATIONS due to Fuzhen Zhang (edt.) for details.