Editing Block matrix (section)

===Inversion{{anchor|Inversion}}===
{{for|more details and derivation using block LDU decomposition|Schur complement}}
{{see also|Helmert–Wolf blocking}}

If a matrix is partitioned into four blocks, it can be [[invertible matrix#Blockwise inversion|inverted blockwise]] as follows:

:<math>{P} = \begin{bmatrix}
    {A} & {B} \\
    {C} & {D}
  \end{bmatrix}^{-1} = \begin{bmatrix}
     {A}^{-1} + {A}^{-1}{B}\left({D} - {CA}^{-1}{B}\right)^{-1}{CA}^{-1} &
      -{A}^{-1}{B}\left({D} - {CA}^{-1}{B}\right)^{-1} \\
    -\left({D}-{CA}^{-1}{B}\right)^{-1}{CA}^{-1} &
       \left({D} - {CA}^{-1}{B}\right)^{-1}
  \end{bmatrix},
</math>

where '''A''' and '''D''' are square blocks of arbitrary size, and '''B''' and '''C''' are [[conformable matrix|conformable]] with them for partitioning. Furthermore, '''A''' and the Schur complement of '''A''' in '''P''': {{nowrap|'''P'''/'''A''' {{=}} '''D''' − '''CA'''{{sup|−1}}'''B'''}} must be invertible.<ref>
{{cite book
  | last = Bernstein
  | first = Dennis
  | title = Matrix Mathematics
  | publisher = Princeton University Press
  | year = 2005
  | pages = 44
  | isbn = 0-691-11802-7
}}</ref>

Equivalently, by permuting the blocks:

:<math>{P} = \begin{bmatrix}
    {A} & {B} \\
    {C} & {D}
  \end{bmatrix}^{-1} = \begin{bmatrix}
     \left({A} - {BD}^{-1}{C}\right)^{-1} &
      -\left({A}-{BD}^{-1}{C}\right)^{-1}{BD}^{-1} \\
    -{D}^{-1}{C}\left({A} - {BD}^{-1}{C}\right)^{-1} &
       \quad {D}^{-1} + {D}^{-1}{C}\left({A} - {BD}^{-1}{C}\right)^{-1}{BD}^{-1}
  \end{bmatrix}.
</math><ref name=":0" />

Here, '''D''' and the Schur complement of '''D''' in '''P''': {{nowrap|'''P'''/'''D''' {{=}} '''A''' − '''BD'''{{sup|−1}}'''C'''}} must be invertible.

If '''A''' and '''D''' are both invertible, then:

: <math>
  \begin{bmatrix}
    {A} & {B} \\
    {C} & {D}
  \end{bmatrix}^{-1} = \begin{bmatrix}
    \left({A} - {B} {D}^{-1} {C}\right)^{-1} & {0} \\
    {0} & \left({D} - {C} {A}^{-1} {B}\right)^{-1}
  \end{bmatrix} \begin{bmatrix}
                     {I} & -{B} {D}^{-1} \\
    -{C} {A}^{-1} &                  {I}
  \end{bmatrix}.
</math>

By the [[Weinstein–Aronszajn identity]], one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.

====Computing submatrix inverses from the full inverse====

By the symmetry between a matrix and its inverse in the block inversion formula, if a matrix '''P''' and its inverse '''P'''<sup>−1</sup> are partitioned conformally:

:<math>P = \begin{bmatrix}
    {A} & {B} \\
    {C} & {D}
  \end{bmatrix}, \quad P^{-1} = \begin{bmatrix}
    {E} & {F} \\
    {G} & {H}
  \end{bmatrix}</math>

then the inverse of any principal submatrix can be computed from the corresponding blocks of '''P'''<sup>−1</sup>:

:<math>{A}^{-1} = {E} - {FH}^{-1}{G}</math>

:<math>{D}^{-1} = {H} - {GE}^{-1}{F}</math>

This relationship follows from recognizing that '''E'''<sup>−1</sup> = '''A''' − '''BD'''<sup>−1</sup>'''C''' (the Schur complement), and applying the same block inversion formula with the roles of '''P''' and '''P'''<sup>−1</sup> reversed.<ref>{{cite web|title=Is this formula for a matrix block inverse in terms of the entire matrix inverse known?|url=https://mathoverflow.net/questions/495299/is-this-formula-for-a-matrix-block-inverse-in-terms-of-the-entire-matrix-inverse|website=MathOverflow}}</ref>
<ref>{{cite journal|last1=Escalante-B.|first1=Alberto N.|last2=Wiskott|first2=Laurenz|title=Improved graph-based SFA: Information preservation complements the slowness principle|journal=Machine Learning|year=2016|volume=|issue=|pages=|doi=10.1007/s10994-016-5563-y|url=https://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1405-55462016000200251}}</ref>