Generalized Bruhat decomposition in commutative domains
aa r X i v : . [ c s . S C ] F e b Generalized Bruhat decomposition incommutative domains
Gennadi Malaschonok ⋆ Tambov State University,Internatsionalnaya 33, 392622 Tambov, Russia [email protected]
Abstract.
Deterministic recursive algorithms for the computation ofgeneralized Bruhat decomposition of the matrix in commutative domainare presented. This method has the same complexity as the algorithm ofmatrix multiplication.
A matrix decomposition of a form A = V wU is called the Bruhat decompositionof the matrix A , if V and U are nonsingular upper triangular matrices and w is a matrix of permutation. It is usually assumed that the matrix A is definedin a certain field. Bruhat decomposition plays an important role in the theoryof algebraic groups. The generalized Bruhat decomposition was introduced anddeveloped by D.Grigoriev[1],[2].In [3] there was constructed a pivot-free matrix decomposition method in acommon case of singular matrices over a field of arbitrary characteristic. Thisalgorithm has the same complexity as matrix multiplication and does not requirepivoting. For singular matrices it allows us to obtain a nonsingular block of thebiggest size.Let R be a commutative domain, F be the field of fractions over R . We wantto obtain a decomposition of matrix A over domain R in the form A = V wU ,where V and U are upper triangular matrices over R and w is a matrix ofpermutation, which is multiplied by some diagonal matrix in the field of fractions F . Moreover each nonzero element of w has the form ( a i a i − ) − , where a i is someminor of order i of matrix A .We call such triangular decomposition the Bruhat decomposition in the com-mutative domain R .In [6] a fast algorithm for adjoint matrix computation was proposed. On thebasis of this algorithm for computing the adjoint matrix in the [8] proposed afast algorithm for LDU decomposition. However, this algorithm is required tocalculate the adjoint matrix and use it to calculate
LDU decomposition. ⋆ Preprint of the paper: G.I.Malaschonok. Generalized Bruhat decomposition incommutative domains / in book: Computer Algebra in Scientific Computing.CASC’2013. LNCS 8136, Springer, Heidelberg, 2013, pp.231-242. Supported by partRussian Foundation for Basic Research No. 12-07-00755a Gennadi Malaschonok
In this paper, we propose another algorithm that does not rely on the cal-culation of the adjoint matrix and which costs less number of operations. Weconstruct the decomposition in the form A = LDU , where L and U are lowerand upper triangular matrices, D is a matrix of permutation, which is multipliedby some diagonal matrix in the field of fractions F and has the same rank asthe matrix A . Then the Bruhat decomposition V wU in the domain R may beeasily obtained using the matrices L , D and U . Let R be a commutative domain, A = ( a i,j ) ∈ R n × n be a matrix of order n , α ki,j be k × k minor of matrix A which disposed in the rows 1 , , . . . , k − , i andcolumns 1 , , . . . , k − , j for all integers i, j, k ∈ { , . . . , n } . We suppose that therow i of the matrix A is situated at the last row of the minor, and the column j of the matrix A is situated at the last column of the minor. We denote α = 1and α k = α kk,k for all diagonal minors (1 ≤ k ≤ n ). And we use the notation δ ij for Kronecker delta.Let k and s be integers in the interval 0 ≤ k < s ≤ n , A ks = ( α k +1 i,j ) be thematrix of minors with size ( s − k ) × ( s − k ) which has elements α k +1 i,j , i, j = k + 1 , . . . , s − , s, and A n = ( α i,j ) = A .We shall use the following identity (see [4], [5]): Theorem 1 (Sylvester determinant identity).
Let k and s be an integers in the interval ≤ k < s ≤ n . Then it is true that det( A ks ) = α s ( α k ) s − k − . (1) Theorem 2 (LDU decomposition of the minors matrix).
Let A = ( a i,j ) ∈ R n × n be the matrix of rank r , α i = 0 for i = k, k + 1 , . . . , r , r ≤ s ≤ n , then the matrix of minors A ks is equal to the following product ofthree matrices: A ks = L ks D ks U ks = ( a ji,j )( δ ij α k ( α i − α i ) − )( a ii,j ) . (2) The matrix L ks = ( a ji,j ) , i = k + 1 . . . s , j = k + 1 . . . r , is a low triangular matrixof size ( s − k ) × ( r − k ) , the matrix U ks = ( a ii,j ) , i = k + 1 . . . r , j = k + 1 . . . s , isan upper triangular matrix of size ( r − k ) × ( s − k ) and D ks = ( δ ij α k ( α i − α i ) − ) , i = k + 1 . . . r , j = k + 1 . . . r , is a diagonal matrix of size ( r − k ) × ( r − k ) .Proof. Let us write the matrix equation (2) for k + 1 = r ( a k +1 i,j ) = ( a k +1 i,k +1 )( δ k +1 ,k +1 a k ( a k a k +1 ) − )( a k +1 k +1 ,j ) (3)This equation is correct due to Sylvester determinant identity a k +1 i,j a k +1 − a k +1 i,k +1 a k +1 k +1 ,j = a k +2 i,j a k , (4) eneralized Bruhat decomposition in commutative domains 3 and the equality a k +2 i,j = 0. This equality is a consequence of the fact that minors a k +2 i,j have the order greater then the rank of the matrix A .Let for all h , k < h < r , the statement (1) be correct for matrices A hs =( a h +1 i,j ). We have to prove it for h = k . Let us write one matrix element in (2)for the matrix A k +1 s = ( a k +2 i,j ) : a k +2 i,j = min ( i,j,r ) X t = k +2 a ti,t α k +1 ( α t − α t ) − a tt,j . We have to prove the corresponding expression for the elements of the matrix A ks . Due to the Sylvester determinant identity (3) we obtain a k +1 i,j = a k +1 i,k +1 ( α k +1 ) − a k +1 k +1 ,j + α k ( α k +1 ) − a k +2 i,j = a k +1 i,k +1 α k ( α k α k +1 ) − a k +1 k +1 ,j + α k ( α k +1 ) − min ( i,j,r ) X t = k +2 a ti,t α k +1 ( α t − α t ) − a tt,j = min ( i,j ) X t = k +1 a ti,t α k ( α t − α t ) − a tt,j . Consequence 1 (LDU decomposition of matrix A ) Let A = ( a i,j ) ∈ R n × n ,be the matrix of rank r , r ≤ n , α i = 0 for i = 1 , , . . . , r , then matrix A is equalto the following product of three matrices: A = L n D n U n = ( a ji,j )( δ ij ( α i − α i ) − )( a ii,j ) . (4) The matrix L n = ( a ji,j ) , i = 1 . . . n , j = 1 . . . r , is a low triangular matrix ofsize n × r , the matrix U n = ( a ii,j ) , i = 1 . . . r , j = 1 . . . n , is an upper triangularmatrix of size r × n and D n = ( δ ij ( α i − α i ) − ) , i = 1 . . . r , j = 1 . . . r , is adiagonal matrix of size r × r . Let I n be the identity matrix and P n be the matrix with second unit diagonal. Consequence 2 (Bruhat decomposition of matrix A ) Let matrix A = ( a i,j ) have the rank r , r ≤ n , and B = P n A . Let B = LDU be the LDU-decompositionof matrix B . Then V = P n LP r and U are upper triangular matrices of size n × r and r × n correspondingly and A = V ( P r D ) U (5) is the Bruhat decomposition of matrix A . We are interested in the block form of decomposition algorithms for LDUand Bruhat decompositions. Let us use some block matrix notations.For any matrix A (or A pq ) we denote by A i ,i j ,j (or A p ; i ,i q ; j ,j ) the block whichstands at the intersection of rows i + 1 , . . . , i and columns j + 1 , . . . , j of thematrix. We denote by A i i the diagonal block A i ,i i ,i . Gennadi Malaschonok
Input: ( A kn , α k ), 0 ≤ k < n . Output: { L kn , { α k +1 , α k +2 , . . . , α n } , U kn , M kn , W kn } ,where D kn = α k diag { α k α k +1 , . . . , α n − α n } − , M kn = α k ( L kn D kn ) − , W kn = α k ( D kn U kn ) − .1. If k = n − A n − n = ( a n ) is a matrix of the first order, then we obtain { a n , { a n } , a n , a n − , a n − } , D n − n = ( α n ) − .
2. If k = n − A n − n = (cid:18) α n − βγ δ (cid:19) is a matrix of second order, then we obtain (cid:26) (cid:18) α n − γ α n (cid:19) , { α n − , α n } , (cid:18) α n − β α n (cid:19) , (cid:18) α n − − γ α n − (cid:19) , (cid:18) α n − − β α n − (cid:19) (cid:27) where α n = ( α n − ) − (cid:12)(cid:12)(cid:12)(cid:12) α n − βγ δ (cid:12)(cid:12)(cid:12)(cid:12) , D n − n = α n − diag { α n − α n − , α n − α n } − .3. If the order of the matrix A kn more than two (0 ≤ k < n − s in the interval ( k < s < n ) and divide the matrix into blocks A kn = (cid:18) A ks BC D (cid:19) . (6)3.1. Recursive step { L ks , { α k +1 , α k +2 , . . . , α s } , U ks , M ks , W ks } = LDU ( A ks , α k )3.2. We compute e U = ( α k ) − M ks B , e L = ( α k ) − C W ks , (7) A sn = ( α k ) − α s ( D − e LD ks e U ) . (8)3.3. Recursive step { L sn , { α s +1 , α s +2 , . . . , α n } , U sn , M sn , W sn } = LDU ( A sn , α s )3.4 Result: { L kn , { α k +1 , α k +2 , . . . , α n } , U kn , M kn , W kn } , where L kn = (cid:18) L ks e L L sn (cid:19) , U kn = (cid:18) U ks e U U sn (cid:19) , (9) M kn = (cid:18) M ks − M sn e LD ks M ks /α k M sn (cid:19) , (10) W kn = (cid:18) W ks − W ks D ks e U W sn /α k W sn (cid:19) . (11) eneralized Bruhat decomposition in commutative domains 5 Proof of the correctness of this algorithm is based on several determinant iden-tities.
Definition 1 ( δ ki,j minors and G k matrices). Let A ∈ R n × n be a matrix. The determinant of the matrix, obtained from theupper left block A ,k ,k of matrix A by the replacement in matrix A of the column i by the column j is denoted by δ ki,j . The matrix of such minors is denoted by G ks = ( δ k +1 i,j ) (12)We need the following theorem (see [4] and [5]): Theorem 3 (Base minor’s identity).
Let A ∈ R n × n be a matrix and i, j, s, k , be integers in the intervals: ≤ k
Let A ∈ R n × n be a matrix and s, k , be integers in the intervals: ≤ k < s ≤ n . Then the following identities are true α s U k ; k +1 ,sn ; s +1 ,n = U ks G k ; k +1 ,sn ; s +1 ,n . (14) α s A k ; k +1 ,sn ; s +1 ,n = A ks G k ; k +1 ,sn ; s +1 ,n . (15)The block A k ; k +1 ,sn ; s +1 ,n of the matrix A kn was denoted by B . Due to Sylvesteridentity we can write the equation for the adjoint matrix( A ks ) ∗ = ( A ks ) − ( α s )( α k ) s − k − (16)Let us multiply both sides of equation (15) by adjoint matrix ( A ks ) ∗ and use theequation (16). Then we get Consequence 4 ( A ks ) ∗ B = ( A ks ) ∗ A k ; k +1 ,sn ; s +1 ,n = ( α k ) s − k − G k ; k +1 ,sn ; s +1 ,n . (17)As well as L ks D ks U ks = A ks , M ks = α k ( L ks D ks ) − = α k U ks ( A ks ) − and W ks = α k ( D ks U ks ) − . (18)Therefor e U = ( α k ) − M ks B = ( α k ) − U ks ( A ks ) − B = ( α s ) − ( α k ) − s + k U ks ( A ks ) ∗ B . (19)Equations (19), (17), (14) give the Gennadi Malaschonok
Consequence 5 e U = U k ; k +1 ,sn ; s +1 ,n (20)In the same way we can prove Consequence 6 e L = L k ; s +1 ,nn ; k +1 ,s . (21)Now we have to prove the identity (8). Due to the equations (14)-(19) weobtain e LD ks e U = ( α k ) − C W ks D ks ( α k ) − M ks B =( α k ) − C ( A ks ) − B = ( α k ) − s + k − ( α s ) − C ( A ks ) ∗ B (22)The identity A sn = ( α k ) − ( α s D − ( α k ) − s + k +1 C ( A ks ) ∗ B ) (23)was proved in [4] and [5]. Due to (20) and (21) we obtain the identity (8).To prove the formula (10) and (11) it is sufficient to verify the identities M kn = α k ( L kn D kn ) − and W kn = α k ( D kn U kn ) − using (9),(10), (11) and definition D kn = α k diag { α k α k +1 , . . . , α n − α n } − . Theorem 4.
The algorithm has the same complexity as matrix multiplication.Proof.
The total amount of matrix multiplications in (7)-(15) is equal to 7 andthe total amount of recursive calls is equal to 2. We do not consider multiplica-tions of the diagonal matrices.We can compute the decomposition of the second order matrix by means of7 multiplicative operations. Therefore we get the following recurrent equality forcomplexity t ( n ) = 2 t ( n/
2) + 7 M ( n/ , t (2) = 7 . Let γ and β be constants, 3 ≥ β >
2, and let M ( n ) = γn β + o ( n β ) be the numberof multiplication operations in one n × n matrix multiplication.After summation from n = 2 k to 2 we obtain7 γ (2 β · ( k − + . . . + 2 k − β · ) + 2 k − γ n β − n β − β − n. Therefore the complexity of the decomposition is ∼ γn β β − eneralized Bruhat decomposition in commutative domains 7 Definition 2.
A decomposition of the matrix A of rank r over a commutativedomain R in the product of five matrices A = P LDU Q (24) is called exact triangular decomposition if P and Q are permutation matrces, L and P LP T are nonsingular lower triangular matrices, U and Q T U Q are non-singular upper triangular matrices over R , D = diag( d − , d − , .., d − r , , .., isa diagonal matrix of rank r , d i ∈ R \{ } , i = 1 , ..r . Designation:
ETD ( A ) = ( P, L, D, U, Q ). Theorem 5 (Main theorem).
Any matrix over a commutative domain hasan exact triangular decomposition.
Before proceeding to the proof, we note that the exact triangular decompo-sition relates the LU decomposition and the Bruhat decomposition in the fieldof fractions.If D matrix is combined with L or U , we get the expression A = P LU Q .This is the LU -decomposition with permutations of rows and columns. If thefactors are grouped in the following way: A = ( P LP T )( P DQ )( Q T U Q ) , then we obtain LDU -decomposition. If S is a permutation matrix in which theunit elements are placed on the secondary diagonal, then ( S L S )( S T D ) U is theBruhat decomposition of the matrix ( SA ).Bruhat decomposition can be obtained from those P LU Q -decompositionthat satisfy the additional conditions: matrix
P LP T and Q T U Q are triangular.Conversely, LU -decomposition can be obtained from the Bruhat decomposition V ′ D ′ U ′ . This can be done if the permutation matrix D can be decomposed intoa product of permutation matrices D ′ = P Q so that the P T L ′ P and QU ′ Q T are triangular matrices.If matrix A is a zero matrix, then ETD ( A ) = ( I, I, , I, I ).If A is a nonzero matrix of the first order, then ETD ( A ) = ( I, a, a − , a, I ).Let us consider a non-zero matrix of order two. We denote A = (cid:18) α βγ δ (cid:19) , ∆ = (cid:12)(cid:12)(cid:12)(cid:12) α βγ δ (cid:12)(cid:12)(cid:12)(cid:12) , ε = (cid:26) ∆, ∆ = 01 , ∆ = 0 . Depending on the location of zero elements, we consider four possible cases. Foreach case, we give the exact triangular decomposition:If α = 0 , then A = (cid:18) α γ ε (cid:19) (cid:18) α − ∆ − α − (cid:19) (cid:18) α β ε (cid:19) . If α = 0 , β = 0 , then A = (cid:18) β δ ε (cid:19) (cid:18) β − − ∆ − β − (cid:19) (cid:18) β ε (cid:19) (cid:18) (cid:19) . Gennadi Malaschonok If α = 0 , γ = 0 , then A = (cid:18) (cid:19) (cid:18) γ ε (cid:19) (cid:18) γ − − ∆ − γ − (cid:19) (cid:18) γ δ ε (cid:19) . If α = β = γ = 0 , δ = 0 , then A = (cid:18) (cid:19) (cid:18) δ
00 1 (cid:19) (cid:18) δ −
00 0 (cid:19) (cid:18) δ
00 1 (cid:19) (cid:18) (cid:19) . There are only two different cases for matrices of size 1 × α = 0 , then (cid:0) α β (cid:1) = (cid:0) α (cid:1) (cid:0) α − (cid:1) (cid:18) α β (cid:19) . If α = 0 , β = 0 , then (cid:0) β (cid:1) = (cid:0) β (cid:1) (cid:0) β − (cid:1) (cid:18) β
00 1 (cid:19) (cid:18) (cid:19) . Two cases for matrices of size 2 × Sentence 1
For all matrices A of size n × m , n, m < there exists an exacttriangular decomposition. In addition, we can formulate the following property, which holds for trian-gular matrices and permutation matrices in the exact triangular decomposition.We denote by I s the identity matrix of order s . Property 1 (Property of the factors).
For a matrix A ∈ R n × m of rank r , r 00 0 (cid:19) has a block d of rank r .Let us denote ( C , C )= C Q T (cid:18) U − − V I (cid:19) and (cid:18) B B (cid:19) = (cid:18) L − − M I (cid:19) P T B .Then for the matrix A we obtain the decomposition: A = (cid:18) P I (cid:19) L M I C d − I d B C D U V d − B I 00 0 I (cid:18) Q I (cid:19) . (26)(2.1) Let B = 0 and C = 0. We can rearrange the block D in the upper leftcorner (cid:18) B C D (cid:19) = (cid:18) II (cid:19) (cid:18) D 00 0 (cid:19) (cid:18) II (cid:19) . Let us find the exact triangular decomposition of D : D = P L D U Q . We denote P = (cid:18) P P (cid:19) I I I , Q = I I I (cid:18) Q Q (cid:19) . Then for the matrix A we obtain the following decomposition: A = P L P T C d − L M I d D 00 0 0 U d − B Q T V U 00 0 I Q . It is easy to check that the decomposition is exact triangular.(2.2) Suppose that at least one of the two blocks of B or C is not zero. Letthe exact triangular decomposition exist for these blocks: C = P L D U Q , B = P L D U Q . We denote P = (cid:18) P I (cid:19) , P = I P 00 0 P , Q = I Q 00 0 Q , Q = (cid:18) Q I (cid:19) , P = P P , Q = Q Q , D ′ = L − P T D Q T U − .Then, basing on the expansion (26) we obtain for the matrix A the decom-position of the form: A = P L P T M L P T C d − L d D D D ′ U V Q T d − B Q T U 00 0 U Q . (27)We denote d and d nondegenerate blocks of the matrices D and D , respec-tively,( V , V ) = V Q T , ( V , V ) = d − B Q T , (cid:18) M M (cid:19) = P T M , (cid:18) M M (cid:19) = P T C d − L = (cid:18) L ′ M I (cid:19) , L = (cid:18) L ′ M I (cid:19) , U = (cid:18) U ′ V I (cid:19) , U = (cid:18) U ′ V I (cid:19) , D ′ = (cid:18) D ′ D ′ D ′ D ′ (cid:19) .M = D ′ d − , V = d − D ′ U ′ , V = d − ( D ′ V + D ′ ) . Then (27) can be written as A = P L M L ′ M M I M L ′ M M I d d 00 0 0 0 00 d D ′ D ′ D ′ D ′ U V V V V U ′ V I U ′ V I Q = P L M L ′ M M I M L ′ M M M I d d 00 0 0 0 00 d D ′ U V V V V U ′ V V V I U ′ V I Q . (28)Find the exact triangular decomposition D ′ : D ′ = P L D U Q , (29)Let us denote the matrices P = diag( I, I, I, I, P ), Q = diag( I, I, I, I, Q ), P = P P , Q = Q Q , ( M ′ , M ′ , M ′ ) = P T ( M , M , M ) ( V ′ , V ′ , V ′ ) =( V , V , V ) Q T .After substituting (29) into (28) we obtain the decomposition of the matrix A as A = P L M L ′ M M I M L ′ M ′ M ′ M ′ L d d 00 0 0 0 00 d D U V V V V ′ U ′ V V V ′ I U ′ V ′ U Q . (30) eneralized Bruhat decomposition in commutative domains 11 We rearrange the blocks d , d and D to obtain the diagonal matrix d =diag( d , d , d , D , P and Q : P = , Q = , P d d 00 0 0 0 00 d D Q = d . As a result, we obtain the decomposition: A = P LdUQ , (31)with permutation matrices P = P P T and Q = Q T Q , diagonal matrix d and triangular matrices L = P L M L ′ M M I M L ′ M ′ M ′ M ′ L P T = L M L ′ M L ′ M ′ M ′ M ′ L M M I U = Q T U V V V V ′ U ′ V V V ′ I U ′ V ′ U Q = U V V V ′ V U ′ V V ′ V U ′ V ′ 00 0 0 U 00 0 0 0 I We show that the expansion (31) is an exact triangular decomposition. To dothis, we must verify that the matrices L = P LP T and Q = Q T UQ aretriangular, and the matrices P , L , U , Q satisfy the properties ( α ) and ( β ).It is easy to see that all matrices in sequence L = P L P T , L = P L P T , L = P L P T , L = P L P T (32)are triangular and L = L .Similarly, all of the matrices in the sequence U = Q T L Q , U = Q T U Q , U = Q T U Q , U = Q T U Q (33)are triangular and U = U . For the matrices L and U Property 1 ( α ) is satisfied. To verify the properties( β ), the unit block in the lower right corner of the matrix L and U should bereplaced by an arbitrary triangular block, respectively, the lower triangle for L and the upper triangular for U . We check that all the matrices in (32) and(33) will be still triangular. This is based on the fact that the exact triangulardecompositions for matrices A , B , C , D ′ have the property ( β ). Algorithms for finding the LDU and Bruhat decomposition in commutative do-main are described. These algorithms have the same complexity as matrix mul-tiplication. − = − 24 0 12 10 60 15 40 0 6 10 0 0 3 / ( − / ( − / 18 0 01 / − − 160 0 0 60