nLab Schur complement

Idea

The Schur complement of the $(i,j)$th entry of a block $2\times 2$ matrix

$$\left( \begin{matrix} A & B \\ C & D \end{matrix} \right)$$

is the following expression involving the inverse matrix of that entry:

• $D- C A^{-1} B$ if the entry is (1,1)

• $C- D B^{-1} A$ if the entry is (1,2)

• $B- A C^{-1} D$ if the entry is (2,1)

• $A- B D^{-1} C$ if the entry is (2,2)

whenever the inverses involved make sense. Typically the case at main diagonal are used compatibly with a standard choice of the shape of triangular matrices in Gauss procedure.

These expressions are related to quasideterminants, Gauss elimination procedure and matrix inverse.

The Schur complement is obtained as one of the entries by performing the block matrix version of row operation to obtain a triangular matrix:

$$\left( \begin{matrix} I & 0 \\ -C A^{-1} & I \end{matrix} \right) \left( \begin{matrix} A & B \\ C & D \end{matrix} \right) = \left( \begin{matrix} A & B \\ 0 & D - C A^{-1} B \end{matrix} \right)$$

If we continue with row operations,

$$\left( \begin{matrix} I & - B (D - C A^{-1} B)^{-1} \\ 0 & I \end{matrix} \right) \left( \begin{matrix} A & B \\ 0 & D - C A^{-1} B \end{matrix} \right) = \left( \begin{matrix} A & 0 \\ 0 & D - C A^{-1} B \end{matrix} \right)$$

we obtain a block diagonal matrix, hence

$$\left( \begin{matrix} A & B \\ C & D \end{matrix} \right)^{-1} = \left(\begin{matrix} A^{-1} & 0 \\ 0 & (D - C A^{-1} B)^{-1} \end{matrix}\right) \left(\begin{matrix} I & - B (D - C A^{-1} B)^{-1} \\ 0 & I \end{matrix} \right) \left(\begin{matrix} I & 0 \\ -C A^{-1} & I \end{matrix}\right)$$

$$\left( \begin{matrix} A & B \\ C & D \end{matrix} \right)^{-1} = \left(\begin{matrix} A^{-1}+A^{-1} B (D - C A^{-1} B)^{-1}C A^{-1} & - A^{-1} B (D - C A^{-1} B)^{-1} \\ -(D - C A^{-1} B)^{-1} C A^{-1} & (D - C A^{-1} B)^{-1} \end{matrix} \right)$$

whenever the inverses of $A$ and of the Schur complement $D - C A^{-1} B$ exist.

There are alternative expressions for those (if certain inverses exist) thanks to the homological identities for quasideterminants or equivalently Woodbury matrix identity (wikipedia). For example, if we start with using the Schur complement $A-B D^{-1}C$ and compare the two expressions for the inverse at the position $(1,1)$ we obtain

$$A^{-1}+A^{-1} B (D - C A^{-1} B)^{-1}C A^{-1} = (A-B D^{-1}C)^{-1}$$

References

• Wikipedia, Schur complement

• Wikipedia, Woodbury matrix identity

• D. H. Carlson, What are Schur complements, anyway?, Linear Algebra Appl. 74 (1986) 257–275 doi90127-8)

category: algebra