An LU-decomposition of a matrix A consists of a lower triangular matrix L, an upper tri-angular matrix U, and a permutation matrix P such that PA = LU. Use the results of problems 6.8, 6.9, and 6.10...

An LU-decomposition of a matrix A consists of a lower triangular matrix L, an upper tri-angular matrix U, and a permutation matrix P such that PA = LU. Use the results of problems 6.8, 6.9, and 6.10 to prove that every nonsingular matrix has an LU-decomposition.

Problems 6.8

Prove that the product of two lower triangular matrices is lower triangular and that the inverse of a nonsingular lower triangular matrix is lower triangular.

Problems 6.9

Prove that the product AB of two square matrices is nonsingular if and only if both A and B are nonsingular.

Problems 6.10

Let

Be a triangular factorization of A; let matrices Q_n, Q_n-1,…….Q₀
be defined by Qn = 1 and

And let L*_k
stand for the matrix Q_kL_kQ^t
_k
 Prove that:

(i) Each Q_k
 is a permutation matrix agreeing with I in the first k rows and columns.

(ii)

(iii) Each L^*
_k
is a lower triangular eta matrix whose eta column is the k the column.