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Let A be a square matrix. An LU decomposition is a decomposition of the form
--> A는 square matrix라고 할때, LU Decomposition은 아래와 같은 형태의 Decomposition이다

where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal. For a matrix, this becomes:
--> L과 U는 동일한 크기의 lower와 Upper 삼각 행렬이다. 이것은 L은 Diagonal위쪽이 전부 0이며, 그리고 U는 diagonal 아래쪽이 모두 0이다. 3 x 3행렬을 가지고 표현하면 아래와 같다.

An LDU decomposition is a decomposition of the form
-> LDU Decomposition은 아래와 같은 형태의 Decomposition이다

where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U are one.
-> D는 Diagonal Matrix이고, L과 U는 Unit 삼각 행렬인데, 이것의 의미는 L과 U의 diagonal의 모든 entry들이 1이다.

An LUP decomposition (also called a LU decomposition with partial pivoting) is a decomposition of the form

where L and U are again lower and upper triangular matrices and P is a permutation matrix, (i.e., a matrix of zeros and ones that has exactly one entry 1 in each row and column.)

-> LUP Decomposition도 LU Decomposition이라 부르는데, L과 U는 Lower, Upper 삼각 행렬을 의미하며 P는 Permutation Matrix이다.

An LU decomposition with full pivoting (Trefethen and Bau) takes the form

출처 : Wikipedia