The QR decomposition of a matrix is a decomposition of a general rectangular matrix, A, defined as:
where Q is orthogonal, R is upper-triangular, and P is a permutation matrix. If A has more rows than columns, the last rows of R are 0; if A has more columns than rows, R is upper-trapezoidal.
A QR decomposition object is constructed from a general rectangular matrix. Once an object exists, there are a variety of member functions that can be used to extract its components and to use the object for various operations.
You can control whether pivoting is done, and you can specify that certain rows of the decomposition should be moved to the front or back, by constructing a QR decomposition server object and using it to construct the decomposition. The technique is the same as using a singular value decomposition server.
A complete orthogonal decomposition augments the QR decomposition from Section 8.3 with an orthogonal matrix Z, as shown in the following equation:
where Z' is the transpose of Z. The matrix Z is used to zero out the last few columns of R in the event that R is upper-trapezoidal rather than upper-triangular. A complete orthogonal decomposition object can be created from either a rectangular matrix or a QR decomposition. Once the object is constructed, you can use various member functions to extract its components and to use it for computation.
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