Page 163 - Matrix Analysis & Applied Linear Algebra

3.10 The LU Factorization 157

3.10.7. A n×n is called a band matrix if a ij = 0 whenever |i − j| >w for
some positive integer w, called the bandwidth. In other words, the
nonzero entries of A are constrained to be in a band of w diagonal lines
above and below the main diagonal. For example, tridiagonal matrices
have bandwidth one, and diagonal matrices have bandwidth zero. If
A is a nonsingular matrix with bandwidth w, and if A has an LU
factorization A = LU, then L inherits the lower band structure of A,
and U inherits the upper band structure in the sense that L has “lower
bandwidth” w, and U has “upper bandwidth” w. Illustrate why this
is true by using a generic 5 × 5 matrix with a bandwidth of w =2.

3.10.8. (a) Construct an example of a nonsingular symmetric matrix that
fails to possess an LU (or LDU) factorization.
(b) Construct an example of a nonsingular symmetric matrix that
has an LU factorization but is not positive definite.

 
1 4 5
3.10.9. (a) Determine the LDU factors for A =  41826  (this is the
31630
same matrix used in Exercise 3.10.1).
(b) Prove that if a matrix has an LDU factorization, then the LDU
factors are uniquely determined.
(c) If A is symmetric and possesses an LDU factorization, explain
T
why it must be given by A = LDL .

 
1 2 3
3.10.10. Explain why A =  2 8 12  is positive definite, and then find the
31227
Cholesky factor R.