LU因数分解

I have three objectives in examining LU factorization:
1. Consider the concepts and applications of this technique.
2. Examine at least one specific algorithm and possibly prove it works.
3. Compare the algorithm with the recurrence used by Professor Kung in his exposition of systolic arrays.

The basic format of LU decomposition is A = LU, where L is a lower triangular matrix and U is an upper triangular matrix. A variant, known as the LUP decomposition, represents A as the product of a lower triangular matrix, an upper triangular matrix, and a permutation matrix P (matrix containing only 0s and 1s, with one 1 in each row and column).
The conditions for the existence of an LU decomposition vary with the type of matrix. An invertible matrix has such a factorization if all leading principal minors are non-zero. A minor of a matrix is a determinant of a smaller submatrix obtained by removing specific rows or columns. I am not sure why such a condition enables the factorization to be derived.
A special type of factorization, the Cholesky decomposition, occurs if the matrix is Hermitian and positive definite. Here A = LL*, where L* is the conjugate transpose of L (i.e. rows and columns are interchanged, and corresponding elements in L and L* are complex conjugates). Note that, while Kung indicates that his matrices are all positive definite, he does not claim that they are Hermitian (equal to their conjugate transpose). Thus, the Cholesky decomposition does not seem to apply to the systolic arrays implementation.
One well-known LU factorization algorithm is the Doolittle algorithm, which seems to be a modified procedure for Gaussian elimination.