Theorem 3.3: If is nonsingular, such that is possible and is only a A P PA=LU P permutation of the rows. In fact, may be found such that for P i>k, Theorem 3.4: Suppose is symmetric. If is possible, then the choice of A A=LU Hence, lkk=ukklik=uki. U=LT. Variants of Gaussian elimination LDU factorization: and are strictly lower and upper triangular and is L U D diagonal. Cholesky: so factor A=AT, A=LLT. is symmetric, but cannot be factored into LU form. Fun example: Definition: is if . A positive definite
Theorem 3.3: If A is nonsingular, P such that PA=LU is possible and P is only a permutation of the rows. In fact, P may be found such that for i>k, Theorem 3.4: Suppose A is symmetric. If A=LU is possible, then the choice of lkk=ukklik=uki. Hence, U=LT. Variants of Gaussian elimination LDU factorization: L and U are strictly lower and upper triangular and D is diagonal. Cholesky: A=AT, so factor A=LLT. Fun example: is symmetric, but cannot be factored into LU form. Definition: A is positive definite if .
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