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The Schur complement arises when performing a block Gaussian elimination on the matrix M.In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows: = [] [] [] = [], where I p denotes a p×p identity matrix.
The Schur complement, however, has condition number only of the order 1/h. For performances, the Schur complement method is combined with preconditioning, at least a diagonal preconditioner . The Neumann–Neumann method and the Neumann–Dirichlet method are the Schur complement method with particular kinds of preconditioners.
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.
where the Schur complement of in the block matrix is defined by Q = D − C A − 1 B {\displaystyle {\begin{matrix}Q=D-CA^{-1}B\end{matrix}}} and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition .
In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury [1] [2] – says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix.
A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is triangular. This is a stronger property than being Hessenberg. Just as in the complex case, a family of commuting real matrices {A i} may be simultaneously brought to quasi-triangular form by an orthogonal matrix.
The matrix ¯ is the Schur complement of Σ 22 in Σ. That is, the equation above is equivalent to inverting the overall covariance matrix, dropping the rows and columns corresponding to the variables being conditioned upon, and inverting back to get the conditional covariance matrix.
A particular even supermatrix is a block matrix of the form = [] Such a matrix is ... is known as the Schur complement of A relative to []. An odd ...