試題詳解

五、Which of the following statements about matrix factorization is/are true?
(A) If a matrix A is positive definite, then A has "an" LU factorization, A = LU, where the diagonal entries of U are positive.
(B) Suppose that a matrix A = QR, where Q is an m x n matrix and R is an n X n matrix. If the columns of A are linearly independent, then R must be invertible.
(C) Any factorization of a matrix A = , with matrices U, V square and positive diagonal entries in the matrix D, is called a singular value decomposition of A.
(D) An n x n matrix A is positive definite if and only if A has "a" Cholesky factorization A = for some invertible upper triangular matrix R whose diagonal entries are all positive.
(E) None of the above are true.