April 2011
Intermediate to advanced
364 pages
10h 8m
English
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20.2 SPECIAL MATRIX STRUCTURES
The following subsections explain some of the matrices that have special structures and are relevant to our discussion here.
20.2.1 Plane Rotation (Givens) Matrix
A 5 × 5 plane rotation (or Givens) matrix Gpq is one that looks like the identity matrix except for elements that lie in the locations pp, pq, qp, and qq. Such a matrix is labeled Gpq. For example, the matrix G42 takes the form
(20.2)
where c = cos θ and s = sin θ. The notation commonly used is that the subscript refers to the element that has the negative sin value, which is element at row 4 and column 2 in our example.
Givens matrix is an orthogonal matrix and we have 
. Premultiplying a matrix A by Gpq modifies only rows p and q. All other rows are left unchanged. The elements in rows p and q become
(20.3)
(20.4)
20.2.2 Banded Matrix
A banded matrix with lower bandwidth p and upper bandwidth q implies that all its nonzero elements lie in the main diagonal, the lower p subdiagonals and the upper q superdiagonals. All other elements are zero, that is, when i > j + p and j > i + q
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