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gsm-78-prev - Chapter 12 Triangular factorization and...

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Chapter 12 Triangular factorization and positive deFnite matrices Half the harm that is done in this world Is due to people who want to feel important. They don’t mean to do harm—but the harm does not interest them. Or they do not see it, or they justify it Because they are absorbed in the endless struggle To think well of themselves. T. S. Elliot, The Cocktail Party This chapter is devoted primarily to positive defnite and semidefnite matrices and related applications. To add perspective, however, it is conve- nient to begin with some general observations on the triangular Factorization oF matrices. In a sense this is not new, because the Formula EPA = U or, equivalently, A = P 1 E 1 U that emerged From the discussion oF Gaussian elimination is almost a tri- angular Factorization. Under appropriate extra assumptions on the matrix A C n × n , the Formula A = P 1 E 1 U holds with P = I n . WARNING: We remind the reader that From now on h u , v i = h u , v i st , the standard inner product, and k u k = k u k 2 For vectors u , v F n , unless indicated otherwise. Correspondingly, k A k = k A k 2 , 2 For matrices A . 239
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240 12. Triangular factorization and positive deFnite matrices 12.1. A detour on triangular factorization The notation A [ j,k ] = a jj ··· a jk . . . . . . a kj a kk for A C n × n and 1 j k n (12.1) will be convenient. Theorem 12.1. Ama tr ix A C n × n admits a factorization of the form A = LDU , (12.2) where L C n × n is a lower triangular matrix with ones on the diagonal, U C n × n is an upper triangular matrix with ones on the diagonal and D C n × n is an invertible diagonal matrix, if and only if the submatrices A [1 ,k ] are invertible for k =1 ,... ,n. (12.3) Moreover, if the conditions in (12.3) are met, then there is only one set of matrices, L , D and U , with the stated properties for which (12.2) holds. Proof. Suppose Frst that the condition (12.3) is in force. Then, upon expressing A = · A 11 A 12 A 21 A 22 ¸ in block form with A 11 C p × p , A 22 C q × q and p + q = n , we can invoke the Frst Schur complement formula A = · I p O A 21 A 1 11 I q ¸· A 11 O OA 22 A 21 A 1 11 A 12 I p A 1 11 A 12 OI q ¸ repeatedly to obtain the asserted factorization formula (12.2). Thus, if A 11 = A [1 ,n 1] ,th en α n = A 22 A 21 A 1 11 A 12 is a nonzero number and the exhibited formula states that A = L n · A [1 ,n 1] O n ¸ U n , where L n C n × n is a lower triangular matrix with ones on the diagonal and U n C n × n is an upper triangular matrix with ones on the diagonal. The next step is to apply the same procedure to the ( n 1) × ( n 1) matrix A [1 ,n 1] . This yields a factorization of the form A [1 ,n 1] = e L n 1 · A [1 ,n 2] O n 1 ¸ e U n 1 ,
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12.1. A detour on triangular factorization 241 where e L n 1 C ( n 1) × ( n 1) is a lower triangular matrix with ones on the diagonal and e U n 1 C ( n 1) × ( n 1) is an upper triangular matrix with ones on the diagonal. Therefore, A = L n · e L n 1 O O 1 ¸ A [1 ,n 2] OO n 1 O α n
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gsm-78-prev - Chapter 12 Triangular factorization and...

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