Editing Moore–Penrose inverse (section)

===Updating the pseudoinverse===
For the cases where {{tmath| A }} has full row or column rank, and the inverse of the correlation matrix ({{tmath| A A^* }} for {{tmath| A }} with full row rank or {{tmath| A^*A }} for full column rank) is already known, the pseudoinverse for matrices related to {{tmath| A }} can be computed by applying the [[Sherman–Morrison–Woodbury formula]] to update the inverse of the correlation matrix, which may need less work. In particular, if the related matrix differs from the original one by only a changed, added or deleted row or column, incremental algorithms exist that exploit the relationship.<ref name="G1992">{{Cite thesis |first= Tino |last=Gramß |title= Worterkennung mit einem künstlichen neuronalen Netzwerk |type=PhD dissertation |publisher= Georg-August-Universität zu Göttingen |year = 1992 | oclc = 841706164 }}</ref><ref name="EMTIYAZ2008">{{cite web |first=Mohammad |last=Emtiyaz |title=Updating Inverse of a Matrix When a Column is Added/Removed |date=February 27, 2008 |url=https://emtiyaz.github.io/Writings/OneColInv.pdf }}</ref>

Similarly, it is possible to update the Cholesky factor when a row or column is added, without creating the inverse of the correlation matrix explicitly. However, updating the pseudoinverse in the general rank-deficient case is much more complicated.<ref>{{cite journal|last=Meyer|first=Carl D. Jr.|title=Generalized inverses and ranks of block matrices|journal=SIAM J. Appl. Math.|volume=25|issue=4|date=1973|pages=597–602|doi=10.1137/0125057}}</ref><ref>{{cite journal|last=Meyer|first=Carl D. Jr.|title=Generalized inversion of modified matrices|journal=SIAM J. Appl. Math.|volume=24|issue=3|date=1973|pages=315–23|doi=10.1137/0124033}}</ref>