nLab Schur complement



The Schur complement of the (i,j)(i,j)th entry of a block 2×22\times 2 matrix

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

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

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

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

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

  • ABD 1CA- 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:

(I 0 CA 1 I)(A B C D)=(A B 0 DCA 1B) \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,

(I B(DCA 1B) 1 0 I)(A B 0 DCA 1B)=(A 0 0 DCA 1B) \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

(A B C D) 1=(A 1 0 0 (DCA 1B) 1)(I B(DCA 1B) 1 0 I)(I 0 CA 1 I)\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)
(A B C D) 1=(A 1+A 1B(DCA 1B) 1CA 1 A 1B(DCA 1B) 1 (DCA 1B) 1CA 1 (DCA 1B) 1) \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 AA and of the Schur complement DCA 1BD - 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 ABD 1CA-B D^{-1}C and compare the two expressions for the inverse at the position (1,1)(1,1) we obtain

A 1+A 1B(DCA 1B) 1CA 1=(ABD 1C) 1 A^{-1}+A^{-1} B (D - C A^{-1} B)^{-1}C A^{-1} = (A-B D^{-1}C)^{-1}


category: algebra

Last revised on May 23, 2024 at 09:15:36. See the history of this page for a list of all contributions to it.