Generalized Principal Pivot Transform

Transcription

1 Generalized Principal Pivot Transform M. Rajesh Kannan and R. B. Bapat Indian Statistical Institute New Delhi, , India Abstract The generalized principal pivot transform is a generalization of the principal pivot transform to the singular case, using the Moore-Penrose inverse. In this article we study some properties of the generalized principal pivot transform. We prove that the Moore-Penrose inverse of a range-symmetric, almost skewsymmetric matrix is almost skew-symmetric. It is shown that the generalized principal pivot transform preserves the rank of the symmetric part of a matrix under some conditions. AMS Subject Classification(2010): 15A09 Keywords. Moore-Penrose inverse, generalized principal pivot transform, generalized Schur complement, range-symmetric matrix, J-Hermitian matrix, almost skewsymmetric matrix. 1

2 1 Introduction Let A be an n n complex matrix partitioned into blocks as where A 11 is invertible. The principal pivot transform of A, with respect to A 11, is defined as ppt(a, A 11 ) = A 1 11 A 1 11 A 12 A 21 A 1 11 (A/A 11 ), where (A/A 11 ) = A 22 A 21 A 1 11 A 12 is the Schur complement of A 11 in A. The principal pivot transform has an interesting history, which is dealt with in detail in[7]. In this article we study the notion of principal pivot transforms for singular matrices (that is to say, the case when A 11 is singular.). In Section 2, we introduce notation and state some preliminary results. In Section 3, we define the generalized principal pivot transform and discuss its properties. In Section 4, first we prove that the Moore-Penrose inverse of a range-symmetric, almost skew-symmetric matrix is almost skew-symmetric. Then we prove that the generalized principal pivot transform preserves the rank of the symmetric part of the matrix under some conditions. As a particular case we get that the principal pivot transform of an almost skew-symmetric matrix is almost skew-symmetric. Our work generalizes some results from [5], [7], [8]. 2 Notation, Definitions and Preliminary Results Let C m n (R m n ) denote the set of all m n matrices over the complex (real) numbers. For A C m n, we denote the adjoint of A, the transpose of A, the range space of A and null space of A by A,A t, R(A) and N(A), respectively. For a given A C m n, the unique matrix X C n m satisfying AXA = A, XAX = X, (AX) = AX and (XA) = XA is called the Moore-Penrose inverse of A and is denoted by A. For a given matrix A C n n, the unique matrix X C n n satisfying AXA = A, XAX = X, and AX = XA is called the group inverse of A and is denoted by A #. If A is nonsingular, then A # = A 1 = A. Unlike the Moore-Penrose inverse, which always exists, the group inverse need not exist for all square matri- 2

3 ces. A well known necessary and sufficient condition for the existence of A # is that rank(a) = rank(a 2 ). For complementary subspaces L and M of C n, the projection (not necessarily orthogonal) of C n on L along M will be denoted by P L,M. If, in addition, L and M are orthogonal then we denote this by P L. Some of the well known properties of A and A # which will be frequently used, are ([1]): R(A ) = R(A ); N(A ) = N(A ); AA = P R(A) ; A A = P R(A ); R(A) = R(A # ); N(A) = N(A # ); AA # = P R(A),N(A). In particular, if x R(A ) then x = A Ax and if x R(A) then x = A # Ax. Definition 2.1. A matrix A C n n (R n n ) is said to be range-hermitian (rangesymmetric) if R(A) = R(A )(R(A) = R(A t )). The following result is known for range-hermitian matrices. Theorem 2.1. [1] Let A C n n. Then the following are equivalent: (a) A is range-hermitian, (b) N(A) = N(A ), (c) A = A #. If A and B are square invertible matrices, then (AB) 1 = B 1 A 1. However, for a generalized inverse this need not be true. The following result presents a characterization for the reverse order law to hold for the case of the Moore-Penrose inverse. Theorem 2.2. [Theorem 1.4.2, [2]] Let A C m n and B C n p. Then (AB) = B A if and only if BB A A and A ABB are Hermitian. Next we recall the definition of generalized Schur complement which generalizes the concept of Schur complement. Definition 2.2. [3] Let A = be a partitioned matrix. Then the generalized Schur complement of A 11 in A is defined as (A/A 11 ) = A 22 A 21 A 11A 12. 3

4 3 Generalized Principal Pivot Transform Generalized principal pivot transform was studied by AR. Meenakshi [6]. In [6], the author studied the relationship between the generalized principal pivot transforms and range-hermitian matrices. In this section first we recall the definition of generalized principal pivot transform. Then we prove some basic properties of the generalized principal pivot transform, including the domain-range exchange property. Finally, we prove a relationship between the generalized principal pivot transform and J- Hermitian matrices. Definition 3.1 (Generalized Principal Pivot transform). Let A = C n n, where A 11 C n 1 n 1 and n 1 n. The generalized principal pivot transform of the matrix A with respect to A 11, denoted by gppt(a, A 11 ) is defined to be A 21 A 11 A 22 A 21 A 11A 12. Remark 3.1. The generalized principal pivot transform with respect to any principal submatrix can be defined in the following way. Let α {1, 2,, n}, ᾱ = {1, 2,, n} \ α and A[α, β] be the submatrix of A whose rows and columns are indexed by α, β {1, 2,, n} respectively. We abbreviate A[α, α] by A[α]. Then the generalized principal pivot transform of A with respect to A[α] is defined by gppt(a, A[α]) = A[α] A[α] A[α, ᾱ] A[ᾱ, α]a[α] A[ᾱ, ᾱ] A[ᾱ, α]a[α] A[α, ᾱ]. The proof of the following results is easy and is omitted. Lemma 3.1. Let A = and N(A 11) N(A 12). Then be a partition of A such that N(A 11 ) N(A 21 ) 4

5 (a) A 11 A 11A 12 = A 12, (b) A 21 A 11A 11 = A 21. The following property is known as the domain-range exchange property of ppt(a, A 11 ). If x = x 1 and y = y 1 partitioned conformally to A, then A x 1 = x 2 y 2 x 2 y 1 if and only if ppt(a, A 11 ) y 1 = x 1. y 2 x 2 y 2 The following theorem is a generalization of the domain-range exchange property to the generalized principal pivot transform. Theorem 3.1. Let A = be a partition of A such that N(A 11 ) N(A 21 ) and N(A 11) N(A 12). Given a pair of vectors x, y C n partitioned as x = x 1,y = y 1 conformally with the partition of A, define u, v C n by x 2 y 2 u 1 = y 2, u 2 = x 2, v 1 = x 1 and v 2 = y 2. Then B = gppt(a, A 11 ) is a matrix with the property that for every x, y C n with x 1 R(A 11), y = Ax if and only if v = Bu. Proof. Let Ax = y and x 1 R(A 11). Then x 1 x 2 = y 1 y 2. Hence A 11 x 1 + A 12 x 2 = y 1 and A 21 x 1 + A 22 x 2 = y 2. Solving the above equations we get x 1 = A 11y 1 A 11A 12 x 2 and y 2 = A 21 A 11y 1 A 21 A 11A 12 x 2 + A 22 x 2. So y 1 = x 1. Thus, if Ax = y then Bu = v. A 21 A 11 A 22 A 21 A 11A 12 x 2 y 2 Conversely, suppose Bu = v. Then A 21 A 11 A 22 A 21 A 11A 12 y 1 x 2 = 5

6 x 1 y 2. Thus A 11y 1 A 11A 12 x 2 = x 1 and A 21 A 11y 1 + A 22 x 2 A 21 A 11A 12 x 2 = y 2. Now, in the first equation premultiplying by A 11 and using (a) of Lemma 3.1 we get y 1 = A 11 x 1 + A 12 x 2. Substituting the value of y 1 in the second equation and using (b) of Lemma 3.1 we get y 2 = A 21 x 1 + A 22 x 2. Thus Ax = y. Now we shall prove some basic properties of generalized principal pivot transform. The next theorem gives a factorization for gppt(a, A 11 ), which is a generalization of Lemma 3.4 of [7]. Theorem 3.2. Let A = be a partition of A. Let T 1 be the matrix obtained from the identity matrix by setting the diagonal block corresponding to A 11 to zero block and T 2 = I T 1. Consider the matrices C 1 = T 2 + T 1 A, C 2 = T 1 + T 2 A. Then gppt(a, A 11 ) = C 1 C 2. Proof. We have C 1 = I 0 and C 2 = 0 I. Thus C 1 C 2 = I 0 0 I = A 21 A 11 A 22 A 21 A 11A 12 = gppt(a, A 11 ). In the next theorem we prove that the Moore-Penrose inverse of gppt(a, A 11 ) is equal to the gppt(a, A 22 ). It generalizes Theorem 3.8 of [7]. Theorem 3.3. Let A = be a partition of A such that N(A 11 ) N(A 21 ) and N(A 11) N(A 12). Then gppt(a, A 11 ) = gppt(a, A 22 ). 6

7 Proof. By Theorem 3.2, we have gppt(a, A 11 ) = C 1 C 2. Also, gppt(a, A 22 ) = A 11 A 12 A 22A 21 A 12 A 22 A 22A 21 A 22 = C 2 C 1. So it is enough to prove (C 1 C 2) = C 2 C 1. Now, By Theorem 2.2 it is enough to verify C 1C 1 C 2C 2 and C 2C 2 C 1C 1 are Hermitian. C 1C 1 C 2C 2 = A 11A 11 + A 21A 21 A 21A 22 A 22A 21 A 22A 22 and C 2C 2 C 1C 1 = A 11A 11 + A 11A 12 A 12A 11 A 11A 12 A 12A 11 A 22A 22. Thus C 1C 1 C 2C 2 and C 2C 2 C 1C 1 are Hermitian and hence the result follows. In the following theorem we show that the generalized principal pivot transform is an involution under some conditions. Theorem 3.4. Let A = be a partition A such that N(A 11 ) N(A 21 ), N(A 11) N(A 12). Suppose B = gppt(a, A 11 ), then gppt(b, B 11 ) = A. Proof. We have B = gppt(a, A 11 ) = A 21 A 11 A 22 A 21 A 11A 12. Now, 7

8 gppt(b, B 11 ) = A 11 A 11 A 11A 12 A 21 A 11A 11 A 22 A 21 A 11A 12 A 21 A 11A 11 A 11A 12 =.[By Lemma3.1] Thus gppt(b, B 11 ) = A. Definition 3.2. [4] Let J = diag(t 1, t 2,, t n ) be a diagonal matrix with t i = ±1 for all i = 1, 2,, n. A matrix A C n n is said to be J-Hermitian, if JA J = A. Let A = be a partitioned matrix such that A 11 C n 1 n 1, A 22 C n 2 n 2 and J = I n I n2 with n 1 +n 2 = n. Then we have the following results. Theorem 3.5. If A is J-Hermitian, then gppt(a, A 11 ) is Hermitian. Conversely, suppose gppt(a, A 11 ) is Hermitian, N(A 11 ) N(A 21 ) and N(A 11) N(A 12), then A is J-Hermitian. Proof. Suppose A is J-Hermitian, then A = A 11 A 21 A 12 A 22. Now, gppt(a, A 11 ) = A 11 A 11A 21 A 12A 11 (A/A 11 ) = A 21 A 11 (A/A 11 ). Thus gppt(a, A 11 ) is Hermitian. Conversely, suppose gppt(a, A 11 ) is Hermitian, N(A 11 ) N(A 21 ) and N(A 11) N(A 12). Now, we have A 11 A 11A 21 A 12A 11 (A/A 11 ) 8 = A 21 A 11 (A/A 11 ). Thus

9 A 11 = A 11, A 11A 21 = A 11A 12 and (A/A 11 ) = (A/A 11 ). But A 11A 21 = A 11A 12 implies A 21 = A 12 (by Lemma3.1) and A 21 = A 12. Also (A/A 11 ) = (A/A 11 ) implies A 22 = A 22. Thus A is J-Hermitian. Theorem 3.6. If A is Hermitian, then gppt(a, A 11 ) is J-Hermitian. Conversely, suppose gppt(a, A 11 ) is J-Hermitian, N(A 11 ) N(A 21 ) and N(A 11) N(A 12), then A is Hermitian. Proof. Similar to the proof of Theorem Almost Skew-symmetric matrices For A R n n, we write A = S(A) + K(A), where S(A) = A+At and K(A) = A At 2 2 are the symmetric and the skew-symmetric part of the matrix A, respectively. Definition 4.1. [5] A matrix A R n n is said to be almost skew-symmetric if the symmetric part S(A) has rank 1. The following theorem is a generalization of Theorem 4.1 of [5]. The proof technique is different. Theorem 4.1. Let A R n n be a range-symmetric matrix. Suppose A is an almost skew-symmetric matrix, then A is also an almost skew-symmetric matrix. Proof. Suppose A is range-symmetric. We shall show that rank(a +A t ) = rank(a+ A t ). We have A t A A = A t AA = A t and A t (A + A t )A = (A + A t ). Thus rank(a + A t ) rank(a + A t ). Now, A t A t (A + A t )AA = A t (A t + A)A. Also A t A t (A + A t )AA = A + A t AA t = A + A t. Thus rank(a + A t ) rank(a + A t ). Hence, if A is almost skew-symmetric, then A is also almost skew-symmetric. In the next theorem we show that the generalized principal pivot transform preserves the rank of the symmetric part S(A) under some conditions. 9

10 Theorem 4.2. Let A = be a partition of A such that N(A 11 ) N(A 21 ) and N(A t 11) N(A t 12). Suppose A 11 is range-symmetric, then rank(s(b)) = rank(s(a)), where B = gppt(a, A 11 ). Proof. Let C 1 = I 0 and C 2 = 0 I. Then B = gppt(a, A 11 ) = C 1 C 2 and C2BC t 2 = A t 11A 11A 11 0 A t 12A 11A 11 + A 21 A 11A 11 A 22. The range symmetry of A 11 along with the conditions N(A 11 ) N(A 21 ) and N(A t 11) N(A t 12) implies A t 11A 11A 11 = A t 11, A t 12A 11A 11 = A t 12 and A 21 A 11A 11 = A 21. Hence C2BC t 2 = A t 11 0 A t 12 + and C2BC t 2 = C2C t 1. Thus C2(B t + B t )C 2 = C t 2C 1 + C t 1C 2 and hence rank(s(b)) rank(s(c t 2C 1 )). Now we shall show that rank(s(b)) rank(s(c t 2C 1 )). We have C t 2(B + B t )C 2 = C t 2C 1 +C t 1C 2. Premultiplying this equation by C t 2 and postmultiplying it by C 2 we get C t 2 C t 2(B + B t )C 2 C 2 = C t 2 (C t 2C 1 + C t 1C 2 )C 2. Now let us prove C t 2 C t 2(B + B t )C 2 C 2 = (B + B t ). It is enough to prove that C 2 C 2C 1 C 2 = C 1 C 2 and C t 2 C t 1C 2 C 2 = C t 2 C t 1. Now C 2 C 2C 1 C 2 = A 21 A 11 (A/A 11 ) = C 1 C 2 and C t 2 C t 1C 2 C t 2 = A t 11 A t 11A 21 A 12A t 11 (A/A 11 ) t = (C 1 C 2) t. Hence B + B t = C t 2 (C t 2C 1 + C t 1C 2 )C 2 and rank(s(c t 2C 1 )) rank(s(b)). Thus rank(s(c t 2C 1 )) = rank(s(b)). Also, C t 2C 1 + C t 1C 2 = A 11 + A t 11 A 12 + A t 21 A 21 + A t 12 A 22 + A t 22 = 2S(A). Hence ranks(b) = 10

11 rank(s(c t 2C 1 )) = ranks(a). Remark 4.1. Theorem 4.2 generalizes Theorem 4.1 of [8]. In [8], the authors have used Sylvester s law of inertia in their proof, but our proof involves only rank arguments. Remark 4.2. The generalized principal pivot transform need not preserve the rank of the matrix. For, consider the matrix A = and A 11 = Then gppt(a, A 11 ) = , which has rank 1. Theorem 4.3. Let A = be a partition of A such that N(A 11 ) N(A 21 ) and N(A t 11) N(A t 12). Suppose A is almost skew-symmetric and A 11 is range-symmetric. Then gppt(a, A 11 ) is almost skew-symmetric. Proof. Similar to Theorem 4.2 Corollary 4.1. Let A = be a partition of A such that N(A 11 ) N(A 21 ) and N(A t 11) N(A t 12). Suppose A is almost skew-symmetric and A 11 is range-symmetric. Then the generalized Schur complement of A 11 in A has rank at most one. Remark 4.3. In general the generalized Schur complement of an almost skew-symmetric matrix may not be almost skew-symmetric. 11

12 For, let A = A is Acknowledgements and A 11 = 1, the generalized Schur complement of A 11 in The first author would like to thank Indian Statistical Institute, Delhi centre for financial support and Prof. AR. Meenakshi for sending a reprint of [6]. The second author acknowledges support from the JC Bose Fellowship, Department of Science and Technology, Government of India. 12

13 References [1] Ben-Israel, A. and T.N.E. Greville, Generalized inverses: theory and applications, Pure and Applied Mathematics, Wiley-Interscience, New York, [2] Campbell, S.L. and C. D. Meyer, Generalized inverses of linear transformations, SIAM, [3] Carlson, David and Haynsworth, Emilie and Markham, Thomas, A generalization of the Schur complement by means of the Moore-Penrose inverse, SIAM J. Appl. Math., 26 (1974), [4] Kresimir Veselic, Damped Oscillations of Linear Systems: A Mathematical Introduction, Springer-Verlag, Berlin, Heidelberg, [5] McDonald, J. J. and Psarrakos, P. J. and Tsatsomeros, M. J., Almost skewsymmetric matrices, Rocky Mountain J. Math., 34 (2004), [6] Meenakshi, AR., Principal pivot transforms of an EP matrix, C. R. Math. Rep. Acad. Sci. Canada, 8(2) (1986), [7] Tsatsomeros, M. J., Principal pivot transforms: properties and applications, Linear Algebra Appl., 307 (2000), [8] Slyusareva, Olga and Tsatsomeros, M. J., Mapping and preserver properties of the principal pivot transform, Linear Multilinear Algebra, 53 (2008),