Nullity theorem
The nullity theorem is a mathematical
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 (Strang & Nguyen 2004):
More generally, if a submatrix is formed from the rows with indices {i1, i2, …, im} and the columns with indices {j1, j2, …, jn}, then the complementary submatrix is formed from the rows with indices {1, 2, …, N} \ {j1, j2, …, jn} and the columns with indices {1, 2, …, N} \ {i1, i2, …, im}, 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, 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, ISSN 0024-3795.
- ISSN 1095-7200.