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Unformatted text preview: = N ; 1 N ; 2 : : : 1: (7.2.9) Several details still remain, including
1. the satisfaction of the initial conditions (7.2.5a,b),
2. the orthogonalization procedure (7.2.7a,b), and
3. the test for a linearly independent basis.
Consider the solution of the under-determined system (7.2.5a,b) rst. The r columns
of U1 (a) are calculated so that they are mutually orthogonal and span the null space of
L. A procedure follows.
1. Reduce LT to upper triangular form using the QR algorithm ( 6], Chapter 3). This
involves nding an m m orthogonal matrix Q such that QLT = T :
0 (7.2.10a) Recall that an orthogonal matrix is one where QT Q = I: (7.2.10b) The matrix T is an l l upper triangular matrix and the zero matrix is r l. Pivoting
can frequently be ignored with orthogonal transformations and we'll assume this
to be the case here.
13 2. Choose U1(a) = QT 0
I (7.2.11) where I is the r r identity matrix and the zero matrix is l r. Thus U1 is the
last r columns of QT . Using (7.2.10) and (7.2.11) we verify that LU1(a) = LQT 0 = TT 0] 0 = 0:
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- Spring '14