Khatri-inverse of factorized structured matrices Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal,
[email protected] M.C.GOUVEIA November 13, 2013
Abstract The definition of a generalized inverse given by Khatri in [4] and a necessary and sufficient condition for its existence [5] is revisited. Some results on particular factorized structured matrices, such as specific rank factorization of Hankel matrices are used to obtain an explicit formula for the Khatri-inverse. This work extends those results to domains and proofs are given by means of different tools. A new necessary and sufficient condition for the existence of the Khatri-inverse of particular products of matrices is proved. An application to Toeplitz, circulant and B´ezout matrices is given.
Keywords: Moore-Penrose; von Neumann regularity; Khatri-inverse; Hankel matrix; integral domain. AMS-Subject Classification: 15A09
1
Introduction
Let R be a commutative ring with unity and let M at(R) be the category of matrices with elements in R. Besides the usual linear algebra notation in the literature, we’ll use the following specific notation. Let A = (aij ) ∈ M at(R) be an m × n matrix. If α = {i1 , . . . , il }, β = {j1 , . . . , jk } are subsets of, respectively, {1, . . . , m} and {1, . . . , n}, by Aαβ = A [i1 , . . . , il |j1 , . . . , jk ] we denote the submatrix of A determined by the rows indexed by α and the columns indexed by β. If α = {1, . . . , m} or if β = {1, . . . , n} we write, respectively, Aβ or Aα . If α = β we shorten the notation to A[i1 , . . . , il ]. The submatrix of A obtained deleting the ith row and the jth column of A is mostly denoted by A[ic |j c ]. The identity matrix of order n is mentioned as In . By Jk we mean the invertible matrix Jk = (δi,k−j+1 ), i, j = 1, . . . , k. By ρ or rankA we denote the 1
determinantal rank of A. A m × n matrix A of rank ρ is said to have a full rank factorization over R if A = EF , where E is m × ρ, F is ρ × n and rankE = rankF = ρ. We let a → a ¯ for an involution on R and A = (aij ) → A∗ = (a¯i j)T for the induced involution on M at(R). If A ∈ M at(R), the Moore-Penrose inverse is defined as the unique matrix X such that (1)AXA = A;
(2)XAX = X;
(3)AX = (AX)∗ ;
(4)XA = (XA)∗ ,
usually denoted by A† . A is said von Neumann regular if the first Moore-Penrose equation is consistent. Then X is said a von Neumann inverse of A. If X is a solution of one, two or three Penrose equations we will call it (i), (i, j) or (i, j, k)- inverse and denote it by A(i) , A(i,j) or A(i,j,k) , We recall the following results in [3]. Definition 1. Let A, M, N ∈ M at(C) be matrices of order m × n, , m × m, n × n, respectively, where M and N are invertible. A n × m matrix X is called the Khatri-inverse of A, with respect to M, N if X satisfies the conditions (1)AXA = A (2)XAX = X (3k )(AX)∗ M = M (AX) (4k )(XA)∗ N = N (XA) It is obvious that the Moore-Penrose inverse is the Khatri-inverse when M, N are identity matrices. So, in the following we will denote X by A(k†) . Theorem 2. Whenever it exists the Khatri-inverse is unique. If R = C, a necessary and sufficient condition for the existence of the Khatri-inverse and an explicit formula of A(k†) are established in [5]. Theorem 3. Let A be an m × n matrix over the complexes with rank ρ and rank factorization A = EF . Then the following conditions are equivalent. (i) A has Khatri-inverse with respect to M, N. (ii) E ∗ M E, F N −1 F ∗ are nonsingular and (E ∗ M E)−1 E ∗ M = (E ∗ M ∗ E)−1 E ∗ M ∗ N −1 F ∗ (F N −1 F ∗ )−1 = N ∗−1 F ∗ (F N ∗−1 F ∗ )−1 . In such case, A(k†) = N ∗−1 F ∗ (F N −1 F ∗ )−1 (E ∗ M E)−1 E ∗ M.
2 2.1
Khatri-inverse of structured matrices Generalized Khatri-inverse
Now we extend definition 1 to matrices over a commutative ring with involution a → a ¯. Definition 4. Let A, M, N ∈ M at(R) be matrices of order m × n, , m × m, n × n, respectively, where M and N are von Neumann regular. A n × m matrix X over R is called the generalized Khatri-inverse of A, with respect to M, N and to the induced involution ∗ if X satisfies the conditions (1) AXA = A
(2) XAX = X
(3k ) (AX)∗ M = M (AX) 2
(4k ) (XA)∗ N = N (XA)
Therefore an extension of theorem 3 can be established. Theorem 5. Let A ∈ M at(R) be a singular matrix with rank ρ and rank factorization A = EF . If E, F are von Neumann regular then the following conditions are equivalent. (i) A has a generalized Khatri-inverse with respect to M, N. (ii) E ∗ M E, F N (1) F ∗ are nonsingular and (EE (1) )∗ M = M EE (1) ,
(1)
(F (1) F )∗ N = N (F (1) F ). In such case, A(k†) = N (1) F ∗ (F N (1) F ∗ )−1 (E ∗ M E)−1 E ∗ M.
(2)
Proof. Since E, F are von Neumann regular then there exist (1, 2)-inverses E (1) , F (1) such that EE (1) , F (1) F are idempotent matrices with rankF F (1) = rankF and rankE (1) E = rankE. Hence E (1) E = Iρ = F F (1) and F (1) E (1) is a (1, 2)-inverse of A. The rest of the proof follows from the definition of A(k†) . Now we consider another particular factorization of A and we give a necessary and sufficient condition for the existence of the generalized Khatri-inverse. Theorem 6. Let A, M, N ∈ M at(R) be von Neumann regular matrices. Let P, Q ∈ M at(R) for which there exist matrices P 0 , Q0 over R such that P 0 P A = A = AQQ0 . Then the generalized Khatri-inverse of B = P AQ exists if and only if A has generalized Khatri-inverse A(k†) with respect to M, N and to the involution ∗. In such case B (k†) = Q0 Ak† P 0 .
(3)
Proof. The fundamental facts to prove are: Q0 A(1) P 0 is a (1)-inverse of B; if A is Khatriinvertible then AQ and P A are also Khatri invertible. Afterwords we just follow definitions. If no factorization of A it is known, we have the following important result. Theorem 7. Let A be a singular matrix. Then the following conditions are equivalent. (i) A has generalized Khatri inverse with respect to M, N. (ii) A∗ M A and AN (1) A∗ are von Neumann regular and A(A∗ M A)(1,2) A∗ M = A(A∗ M A)(1, 2)A∗ M ∗ ,
(4)
N (1) A∗ (AN (1) A∗ )(1,2) A = (N ∗ )(1) A∗ (A(N ∗ )(1) A∗ )(1,2) A are satisfied for any choice of (1, 2)-inverse. In such case A(k†) = N (1) A∗ (AN (1) A∗ )(1,2) A(A∗ M A)(1,2) A∗ M.
(5)
Proof. If (ii) is satisfied then, by definition, an easy computation shows that (5) is the generalized Khatri-inverse of A. If (i) is satisfied, then by theorem 5, we have the equalities in (1) which imply (4), since the equations (4) are invariant under the choice of the generalized inverse.
3
2.2
Khatri-inverse of Hankel-related matrices
Let R be an integral domain and let H = (hi+j ), 1 ≤ i ≤ m; 1 ≤ j ≤ n be a Hankel matrix over R with rank ρ. Let r be the r-characteristic of H and k := ρ − r. Let the (r, k)-characteristic of H be defined as in [3]. Therefore we have Theorem 8. Let H ∈ M at(R) be a m × n Hankel matrix such that rankH = ρ, and r(H) = r, 0 < r ≤ ρ < m ≤ n. Then H has a Frobenius full rank factorization in M at(R), α Ir O α H H H = EF := X O = Hτ , (6) Hω O Ik Hω α = {1, . . . , r}, τ = {r + 1, . . . , m − k}, ω = {m − k + 1, . . . , m} for k 6= 0, if and only if the set of the first r rows of H is a basis for the R-module generated by the first m − k rows. What the theorem says is that the factorization (6) exists if and only if one can define X ∈ M at(R) such that XH α = H τ is consistent, or yet, if and only if there is a (1)-inverse of H α such that H τ (H α )(1) H α = H τ . Hence, from theorem 2 in [3], if R is a field this rank factorization always exists and it is unique. If R is a commutative ring with no zero divisors, in case it exists it is unique. Theorem 9. Let H ∈ M at(R) be a singular Hankel matrix with (r, k)-characteristic .Then the following are equivalent. (i) H has a Frobenius full rank factorization . (ii) H has a Khatri-inverse with respect to M, N. (iii) E ∗ M E, F N (1) F ∗ are invertible and (E ∗ M E)−1 E ∗ M = (E ∗ M ∗ E)−1 E ∗ M ∗ ,
(7)
N (1) F ∗ (F N (1) F ∗ )−1 = N ∗(1) F ∗ (F (N (1) )∗ F ∗ )−1 . In such case, H (k†) = N (1) F ∗ (F N (1) F ∗ )−1 (E ∗ M E)−1 E ∗ M.
(8)
Proof. That (i) ⇒ (ii) and (ii) ⇒ (iii) is clearly a consequence of theorem 5 and 7. So it remains to prove that (iii) ⇒ (i), taking in count theorem 8. Consider now a m×n Toeplitz matrix T ∈ M at(C), T = (ti−j ), 0 ≤ i ≤ m−1; 0 ≤ j ≤ n−1. It is clear from the definition that there exist m × n Hankel matrices H, H 0 such that T = Jm H = H 0 Jn .
(9)
Since Jk , k = m, n are invertible necessarily rankT = rankH and, since the Frobenius rank factorization is always defined over the complexes, then T = Jm EF = GF,
E 0 F 0 J n = E 0 G0
(10)
are full rank factorizations of Tm×n . Then the formula of the Khatri-inverse of a Toeplitz matrix is given by theorem 3. If R is an integral domain, (10) are Frobenius rank factorizations of T over the field of fractions of R. So T k† is given by (2) over R if and only if E ∗ M E, F N (1) F ∗ are units. 4
Another option is to apply theorem 6 to (9). Obviously, T = P HQ with P = J, P 0 = J and Q, Q0 the identity matrices. Then the generalized Khatri-inverse of T exists if and only if H k† exists and in this case T (k†) = H k† J. Let C ∈ M at(C) be a circulant matrix. It is known [2] that C = circ(a0 , a1 , . . . , an−1 ) is a n × n circulant matrix (cij ) such that cij := aj−i(modn) . Moreover, C = Un∗ diag(λ1 , . . . λn )Un with Un = √1n Fn is a unitary matrix, where Fn is the n×n Fourier matrix (e−2jkπi/n ), 0 ≤ j, k < n, and λ1 , . . . λn are the eigenvalues of C. Therefore, applying theorem 6 to this factorization, the Khatri-inverse of C exists if and only if (diag(λ1 , . . . λn ))k† exists with respect to M, N . In such case, C (k†) = Un (diag(λ1 , . . . λn ))k† Un∗ . Finally, we remark that the foregoing reasoning can be extended to other Hankel-related matrices such as Bezoutian and resultant matrices. Acknowledgement Thanks are due to the sponsor of this project: Instituto de Telecomunica¸c˜ oes - P´ olo de Coimbra, Portugal: PEst-OE/EEI/LA0008/2013.
References [1] R. B. Bapat, Bhaskara Rao and Manjunata Prasad K., Generalized inverses over integral domains, Linear Algebra and its Applications 165 (1992), 59-69 . [2] P. J. Davis, Circulant Matrices, John Wiley and Sons, Inc. N. Y., 1979. [3] M. C. Gouveia, Regular Hankel Matrices Over Integral Domains, Linear Algebra and its Applications 255 (1997), 335-347. [4] C. G. Khatri, A note on a commutative g-inverse, Sankhy˜ a A, 328 (1970) 299-319. [5] K. M. Prasad, R.B. Bapat, A note on the Khatri-inverse, The Indian Journal of statistics, vol. 54, series A, Pt 2, (1992) 291-295.
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