G.4 LU, LDU, AND CHOLESKY DECOMPOSITIONS
Before describing the Cholesky decomposition for a PD matrix, we summarize other matrix decompositions, given without proofs.
Definition: Leading Principal Submatrix The nth leading principal submatrix 
is a square matrix generated from matrix 
by retaining only the first n rows and first n columns.
Of course, the previous definition assumes 
.
Theorem G.3 (LU decomposition). If all leading principal submatrices An for 
of 
are nonsingular, then
(G.28) 
where 
is a lower triangular matrix with ones along the main diagonal, and 
is an upper triangular matrix.
Note that L is always square whereas U is rectangular if A is not square. A lower (upper) triangular ...
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