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:
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"1, "i"2, …, "i""m"} and the columns with indices {"j"1, "j"2, …, "j""n"}, then the complementary submatrix is formed from the rows with indices {1, 2, …, "N"} \ {"j"1, "j"2, …, "j""n"} and the columns with indices {1, 2, …, "N"} \ {"i"1, "i"2, …, "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.