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:
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 :
More generally, if a submatrix is formed from the rows with indices {i<sub>1</sub>, i<sub>2</sub>, â¦, i<sub>m</sub>} and the columns with indices {j<sub>1</sub>, j<sub>2</sub>, â¦, j<sub>n</sub>}, then the complementary submatrix is formed from the rows with indices {1, 2, â¦, N} \ {j<sub>1</sub>, j<sub>2</sub>, â¦, j<sub>n</sub>} and the columns with indices {1, 2, â¦, N} \ {i<sub>1</sub>, i<sub>2</sub>, â¦, i<sub>m</sub>}, 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.