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 {{harvtxt|Gustafson|1984}}, and for matrices by {{harv|Fiedler|Markham|1986}}.

Partition a matrix and its inverse in four submatrices:

:

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 {{harv|Strang|Nguyen|2004}}:

:$\backslash begin\{align\}$

\operatorname{nullity} \, A &= \operatorname{nullity} \, H, \\

\operatorname{nullity} \, B &= \operatorname{nullity} \, F, \\

\operatorname{nullity} \, C &= \operatorname{nullity} \, G, \\

\operatorname{nullity} \, D &= \operatorname{nullity} \, E.

\end{align}

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.