The Schur complements of a block matrix are functions of its blocks that allow us to derive several useful formulae for the inversion and the factorization of the block matrix itself.

Let us start with a definition.

Definition Let be a block matrixsuch that its blocks and are square matrices. If is invertible, the Schur complement of in isIf is invertible, the Schur complement of in is

Note that the definition does not specify the dimensions of and its blocks. The only essential thing that we need to specify is that and must be square matrices. Everything else follows from the latter requirement.

In fact, suppose that is and is . Then, must be and must be . As a consequence, the product is well-defined and its dimension is . Similarly, the product is well-defined and its dimension is .

Schur complements play a key role in the inversion of block matrices.

Proposition Let be a block matrixsuch that its blocks and are square matrices. If and its Schur complement are invertible, then is invertible and

Proof

The rigorous way to prove the proposition is to multiply by using the rule for the multiplication of block matrices and to show that the result is the identity matrix. However, we take an approach that shows how the formulae for the blocks of have been derived. Suppose that exists and partition it into blocks as follows:where has the same dimension as , and has the same dimension as . By the definition of inverse, should satisfyorwhere is the identity matrix having the same dimension as , and is the identity matrix having the same dimension as . This is equivalent to saying that the four blocks should satisfy the four equationsIf is invertible, we can transform the first equation intoThen we substitute equation (5) into (3) and getorThus, the Schur complement of has already appeared. Let us write it explicitly:If the Schur complement of is invertible, then we haveThus, we have found the -th block of . We can now plug equation (6) into (5) and obtain another block of :The next step is to transform equation (2) as follows:Then, we plug equation (7) into (4):From the latter equation, we obtain one more block of :We can now substitute equation (8) into (7) and recover the last block of :

There is an analogous proposition for the Schur complement of . You may try to prove it as an exercise and then use the proof below to check your solution.

Proof

As before, suppose that exists and partition it into blocks:where has the same dimension as , and has the same dimension as . By the definition of inverse, should satisfyorwhere is the identity matrix having the same dimension as , and is the identity matrix having the same dimension as . Thus, the four blocks need to satisfy the equationsSince is invertible, then equation (4) can be written asBy plugging equation (5) into (2), we obtainorBy using the appropriate notation for the Schur complement of , we can writeSince the Schur complement of is invertible, we can solve the latter equation and get our first solution for a block of :Then we substitute equation (6) into (5) and obtain another block of :We now manipulate equation (3):and substitute (7) into (1):Thus, we derive one more block of :Finally, the last block of is obtained by plugging equation (8) into (7):

The Schur complements are often used to factorize a block matrix into a product of simpler block matrices.

Proposition Let be a block matrixsuch that its blocks and are square matrices. If is invertible, thenwhere are identity matrices and are matrices of zeros.

Proof

Remember that the Schur complement of in isTherefore, the product of the three matrices is

Proof

Remember that the Schur complement of in isTherefore, the product of the three matrices is

When the Schur complements are invertible, they can be used to derive useful factorizations of the inverse of a block matrix.

Proposition Let be a block matrixsuch that its blocks and are square matrices. If and its Schur complement are invertible, thenwhere are identity matrices and are matrices of zeros.

Proof

If we multiply the factorization of into three matrices derived above by the factorization of proposed here, we obtain the identity matrix because

Proof

Below you can find some exercises with explained solutions.

Let be a block matrix

Suppose that and are invertible and and are zero. Invert by using the Schur complement of .

Solution

The Schur complement of isand its inverse isTherefore,

Please cite as:

Taboga, Marco (2021). "Schur complement", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/Schur-complement.

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