Consider the matrix a. Find lower triangular elementary matrices E1, E2, . . ....
Consider the matrix
a. Find lower triangular elementary matrices E1, E2, . . . , Em such that the product
Em · · · E2E1A
is an upper triangular matrix U. Hint: Use elementary row operations to eliminate the entries below the diagonal of A.
b. Find lower triangular elementary matrices M1, M2, . . . , Mm and an upper triangular matrix U such that
A = M1M2 · · · MmU.
c. Find a lower triangular matrix L and an upper triangular matrix U such that
A = LU.
Such a representation of an invertible matrix is called an LU-factorization. The method outlined in this exercise to find an LU-factorization can be streamlined somewhat, but we have seen the major ideas. An LU-factorization (as introduced here) does not always exist. See Exercise 92.
d. Find a lower triangular matrix L with 1’s on the diagonal, an upper triangular matrix U with 1’s on the diagonal, and a diagonal matrix D such that A = LDU. Such a representation of an invertible matrix is called an LDU-factorization.
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