In this exercise and the next we develop a method for determining whether a given symmetric matrix is positive definite. Given an n×n matrix A let A k denote the principal submatrix made up of the...

In this exercise and the next we develop a method for determining whether a given symmetric matrix is positive definite. Given an n×n matrix A let A_k
denote the principal submatrix made up of the first k rows and columns. Show (by induction) that if the first n−1 principal submatrices are nonsingular, then there is a unique lower triangular matrix L with unit diagonal and a unique upper triangular matrix U such that A = LU. (See Appendix C.)