Block LDU decomposition
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Block Cholesky decomposition
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Consider a block matrix:
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where the matrix is assumed to be non-singular,
is an identity matrix with proper dimension, and is a matrix whose elements are all zero.
We can also rewrite the above equation using the half matrices:
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where the Schur complement of
in the block matrix is defined by
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and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition.
The half matrices satisfy that
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Thus, we have
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where
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The matrix can be decomposed in an algebraic manner into
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