By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix " L " and an upper triangular matrix " U " if and only if all its leading principal minors are non-zero.
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:* \ Theta _ i = \ Phi _ i P, where P is a lower triangular matrix obtained by a Cholesky decomposition of \ Sigma _ u such that \ Sigma _ u = PP', where \ Sigma _ u is the covariance matrix of the errors u _ t
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The variable " L " ( standing for lower or left ) is commonly used to represent a lower triangular matrix, while the variable " U " ( standing for upper ) or " R " ( standing for right ) is commonly used for upper triangular matrix.
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