Editing Schur complement (section)

== Properties ==

* If ''p'' and ''q'' are both 1 (i.e., ''A'', ''B'', ''C'' and ''D'' are all scalars), we get the familiar formula for the inverse of a 2-by-2 matrix:
: <math> M^{-1} = \frac{1}{AD-BC} \left[ \begin{matrix} D & -B \\ -C & A \end{matrix}\right] </math>
: provided that ''AD''&nbsp;&minus;&nbsp;''BC'' is non-zero.
* In general, if ''A'' is invertible, then
: <math>\begin{align}
  M &= \begin{bmatrix} A&B\\C&D \end{bmatrix} = 
  \begin{bmatrix} I_p & 0 \\ CA^{-1} & I_q \end{bmatrix}\begin{bmatrix} A & 0 \\ 0 & D - CA^{-1}B \end{bmatrix}\begin{bmatrix} I_p & A^{-1}B \\ 0 & I_q \end{bmatrix}, \\[4pt]
  M^{-1} &= \begin{bmatrix} A^{-1} + A^{-1} B (M/A)^{-1} C A^{-1} & - A^{-1} B (M/A)^{-1} \\ - (M/A)^{-1} CA^{-1} & (M/A)^{-1} \end{bmatrix} 
\end{align}</math>
: whenever this inverse exists.
* (Schur's formula) When ''A'', respectively ''D'', is invertible, the determinant of ''M'' is also clearly seen to be given by
: <math>\det(M) = \det(A) \det\left(D - CA^{-1} B\right)</math>, respectively
: <math>\det(M) = \det(D) \det\left(A - BD^{-1} C\right)</math>,
: which generalizes the determinant formula for 2 × 2 matrices.
* (Guttman rank additivity formula) If ''D'' is invertible, then the [[Rank (linear algebra)|rank]] of ''M'' is given by
: <math> \operatorname{rank}(M) = \operatorname{rank}(D) + \operatorname{rank}\left(A - BD^{-1} C\right)</math>
* ([[Haynsworth inertia additivity formula]]) If ''A'' is invertible, then the ''inertia'' of the block matrix ''M'' is equal to the inertia of ''A'' plus the inertia of ''M''/''A''.
* (Quotient identity) <math>A/B = ((A/C)/(B/C))</math>.<ref>{{Cite journal |last1=Crabtree |first1=Douglas E. |last2=Haynsworth |first2=Emilie V. |date=1969 |title=An identity for the Schur complement of a matrix |url=https://www.ams.org/proc/1969-022-02/S0002-9939-1969-0255573-1/ |journal=Proceedings of the American Mathematical Society |language=en |volume=22 |issue=2 |pages=364–366 |doi=10.1090/S0002-9939-1969-0255573-1 |s2cid=122868483 |issn=0002-9939|doi-access=free }}</ref>
* The Schur complement of a [[Laplacian matrix]] is also a Laplacian matrix.<ref>{{Cite journal |last=Devriendt |first=Karel |date=2022 |title=Effective resistance is more than distance: Laplacians, Simplices and the Schur complement |url=https://linkinghub.elsevier.com/retrieve/pii/S0024379522000039 |journal=Linear Algebra and Its Applications |language=en |volume=639 |pages=24–49 |doi=10.1016/j.laa.2022.01.002|arxiv=2010.04521 |s2cid=222272289 }}</ref>