nullity theorem

The nullity theorem is a mathematical theorem about the inverse of a partitioned matrix, which states that the nullity of a block in a matrix equals the nullity of the complementary block in its inverse matrix. Here, the nullity is the dimension of the kernel. The theorem was proven in an abstract setting by , and for matrices by .

Partition a matrix and its inverse in four submatrices:
:$\begin A & B \\ C & D \end^ = \begin E & F \\ G & H \end.$
The partition on the right-hand side should be the transpose of the partition on the left-hand side, in the sense that if ''A'' is an ''m''-by-''n'' block then ''E'' should be an ''n''-by-''m'' block.

The statement of the nullity theorem is now that the nullities of the blocks on the right equal the nullities of the blocks on the left :
:$\begin \operatorname \, A &= \operatorname \, H, \\ \operatorname \, B &= \operatorname \, F, \\ \operatorname \, C &= \operatorname \, G, \\ \operatorname \, D &= \operatorname \, E. \end$

More generally, if a submatrix is formed from the rows with indices and the columns with indices , then the complementary submatrix is formed from the rows with indices \ and the columns with indices \ , where ''N'' is the size of the whole matrix. The nullity theorem states that the nullity of any submatrix equals the nullity of the complementary submatrix of the inverse.

References

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* {{Citation , last1=Strang , first1=Gilbert , author1-link=Gilbert Strang , last2=Nguyen , first2=Tri , title=The interplay of ranks of submatrices , doi=10.1137/S0036144503434381 , year=2004 , journal=SIAM Review , issn=1095-7200 , volume=46 , issue=4 , pages=637–646, url=http://dspace.mit.edu/bitstream/1721.1/3885/2/HPCES009.pdf , hdl=1721.1/3885 , hdl-access=free .

Matrix theory