In the previous post, two proofs were given for the Cauchy-Schwarz inequality. We will now consider another proof.
The Schur complement probably goes back to Carl Friedrich Gauss (1777-1855) (for Gaussian elimination). To solve the linear system
by mimicking Gaussian elimination, that is, if is square and nonsingular, then by eliminating , by multiplying the first equation by and subtracting the second equation, we get
Note, the matrix is the Schur complement of in , and if it is square and nonsingular, then we can obtain the solution to our system.
The Schur complement comes up in Issai Schur’s (1875-1941) seminal lemma published in 1917, in which the Schur determinate formula was introduced. By considering elementary operations of partitioned matrices, let
where is square and nonsingular. We can change so that the lower-left and upper-right submatrices become . More precisely, we can make the lower-left and upper-right submatrices by subtracting the first row multiplied by from the second row, and by subtracting the first column multiplied by from the second column. In symbols,
and in equation form,
Note that we have obtain the following factorization of :
By taking the determinants
we obtain the Schur’s determinant formula for block matrices,
Mathematician Emilie Virginia Haynsworth (1916-1985) introduced a name and a notation for the Schur complement of a square nonsingular (or invertible) submatrix in a partitioned (two-way block) matrix. The term Schur complement first appeared in Emily’s 1968 paper On the Schur Complement in Basel Mathematical Notes, then in Linear Algebra and its Applications Vol. 1 (1968), AMS Proceedings (1969), and in Linear Algebra and its Applications Vol. 3 (1970).
We will now present a block matrix proof, focusing on complex matrices.
Proof: Let . Then
By taking the Schur complement of , we arrive at
which ensures, when and are square, that
Equality occurs if and only if rank rank ; that is, by the Guttman rank additivity formula, rank rank rank if and only if . When is nonsingular, is nonsingular if and only if is nonsingular.
 Zhang, Fuzhen. Matrix theory: basic results and techniques. Springer Science & Business Media, 2011.