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:

${\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]}^{-1}=\left[\begin{array}{cc}E& F\\ G& H\end{array}\right].$

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 :

$\begin{array}{cc}\mathrm{nullity}A& =\mathrm{nullity}H,\\ \mathrm{nullity}B& =\mathrm{nullity}F,\\ \mathrm{nullity}C& =\mathrm{nullity}G,\\ \mathrm{nullity}D& =\mathrm{nullity}E.\end{array}$

More generally, if a submatrix is formed from the rows with indices {i₁, i₂, …, i_m} and the columns with indices {j₁, j₂, …, j_n}, then the complementary submatrix is formed from the rows with indices {1, 2, …, N} \ {j₁, j₂, …, j_n} and the columns with indices {1, 2, …, N} \ {i₁, i₂, …, i_m}, 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

• Gustafson, William H. (1984), "A note on matrix inversion", Linear Algebra and Its Applications, 57: 71–73, doi:10.1016/0024-3795(84)90177-0, ISSN 0024-3795.
• Fiedler, Miroslav; Markham, Thomas L. (1986), "Completing a matrix when certain entries of its inverse are specified", Linear Algebra and Its Applications, 74 (1–3): 225–237, doi:10.1016/0024-3795(86)90125-4, ISSN 0024-3795.
• Strang, Gilbert; Nguyen, Tri (2004), "The interplay of ranks of submatrices" (PDF), SIAM Review, 46 (4): 637–646, Bibcode:2004SIAMR..46..637S, doi:10.1137/S0036144503434381, hdl:1721.1/3885, ISSN 1095-7200.