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: : \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} E & F \\ G & H \end{bmatrix}. 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{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.…

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: \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} E & F \\ G & H \end{bmatrix}. 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{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 {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.

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