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Text version of the page
| Factorizations | |||||||||||||||||
| Factorizations | |||||||||||||||||
| In this section... | |||||||||||||||||
| "Introduction" on page 1-29 "Cholesky Factorization" on page 1-29 "LU Factorization" on page 1-31 "QR Factorization" on page 1-32 | |||||||||||||||||
| Introduction All three of the matrix factorizations discussed in this section make use of triangular matrices, where all the elements either above or below the diagonal are zero. Systems of linear equations involving triangular matrices are easily and quickly solved using either forward or back substitution. | |||||||||||||||||
| Cholesky Factorization The Cholesky factorization expresses a symmetric matrix as the product of a triangular matrix and its transpose | |||||||||||||||||
| where R is an upper triangular matrix. Not all symmetric matrices can be factored in this way; the matrices that have suchafactorizationaresaidtobepositivedefinite. Thisimpliesthatallthe diagonal elements of A are positive and that the offdiagonal elements are "not too big." The Pascal matrices provide an interesting example. Throughout this chapter, the example matrix A has been the 3-by-3 Pascal matrix. Temporarily switch to the 6-by-6: A = pascal(6) A = 111111 1 2 3 4 5 6 1 3 6 10 15 21 1 4 10 20 35 56 | |||||||||||||||||
| 1-29 | |||||||||||||||||