In this exercise we will examine which invertible n x n matrices A admit an

Chapter 2, Problem 93

(choose chapter or problem)

In this exercise we will examine which invertible n x n matrices A admit an LU-factorization A = LU, as discussed in Exercise 90. The following definition will be useful: For m = 1 ,..., n the principal submatrix A^ of A is obtained by omitting all rows and columns of A past the mth. For example, the matrixA =1 2 3 4 5 6 7 8 7has the principal submatrices'l 2 4 5l 2 3'A(1) = [1], A(2) =IIII rn4 5 6--j007We will show that an invertible n x n matrix A admits an LU-factorization A = LU if (and only if) all its principal submatrices are invertible. a. Let A = LU be an LU-factorization of an n x n matrix A. Use block matrices to show that A(m) = L(m)fj(m) for m = t........b. Use part (a) to show that if an invertible n x n matrix A has an LU-factorization, then all its principal submatrices A^ are invertible. c. Consider an n x n matrix A whose principal submatrices are all invertible. Show that A admits an LU -factorization. (Hint: By induction, you can assume that A ^-1) has an LU-factorization AC77 i) = L'U'.) Use block matrices to find an LU- factorization for A. Alternatively, you can explain this result in terms of Gauss-Jordan elimination (ifall principal submatrices are invertible, then no row swaps are required).