FURTHER RESULTS ON THE PERTURBATION ESTIMATIONS FOR THE DRAZIN INVERSE

. For n × n complex singular matrix A with ind( A ) = k > 1, let A D be the Drazin inverse of A . If a matrix B = A + E with ind( B ) = 1 is said to be an acute perturbation of A , if (cid:107) E (cid:107) is small and the spectral radius of B g B − A D A satisﬁes , where B g is the group inverse of B . The acute perturbation coincides with the stable perturbation of the group inverse, if the matrix B satisﬁes geometrical condition: R B , ( 0 } which introduced by V´elez-Cerrada, Robles, and Castro-Gonz´alez, (Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput. , 215 (2009), 2154–2161). Furthermore, two examples are provided to illustrate the acute perturbation of the Drazin inverse. We prove the correctness of the conjecture in a special case of ind( B ) = 1 by Wei ( Acute perturbation of the group inverse, Linear Algebra Appl. , 534 (2017), 135–157).

In this paper, C m×n is the set of m × n complex matrices. If m = n, then the identity matrix of order n and the null matrix in C n×n are denoted simply by I n and O, respectively. For A ∈ C n×n , we denote R(A) for its range and N (A) for its null space. A * is the conjugate transpose of the matrix A. · denotes the spectral norm. The Drazin inverse of A ∈ C n×n is the unique matrix A D ∈ C n×n satisfying three equations [1,4] where k is the smallest nonnegative integer satisfying rank(A k+1 ) = rank(A k ), k is called the Drazin index of A and is denoted by ind(A). Clearly, ind(A) = 0 if and only if A is nonsingular. If ind(A) = 1, then the Drazin inverse is called the group inverse of A and presented by A g . Let A π = I n − AA D = P N (A k ),R(A k ) be the spectral projector on R( Let A be a singular matrix with ind(A) = k > 1, and B = A+E be a perturbation of A with ind(B) = 1. Campbell and Meyer [3] presented a necessary and sufficient condition for the continuity of the Drazin inverse, In the general case of the perturbation analysis of the group inverse B g and the spectral projector B π (= I n − BB g ), a condition of (C 1 ) is introduced in [6,25] for the stable perturbation of A.
then B is called the stable perturbation of A.
One formula for B π is given in [6,Theorem 4.4] under the condition (C 1 ) is satisfied. It is not easy to judge that B π − A π < 1 with respect to 2-norm.
Wedin [26] and Stewart [21] presented the acute perturbation of the Moore-Penrose inverse A ∈ C m×n in 1970s, respectively. They say that the range spaces Similarly, the range spaces R(A * ) and R(B * ) are acute, if B † B − A † A < 1. The matrices A and B are called acute, if R(A) and R(B) are acute and R(A * ) and R(B * ) are acute. In this case, they say that B is an acute perturbation of A [21,26].
Recently, we extend the acute perturbation from the Moore-Penrose inverse to the weighted Moore-Penrose inverse [17].
This note is organized as follows. In Section 2, we introduce the definition of the acute perturbation of the Drazin inverse, which is equivalent to the condition (C 1 ) for matrices [6], and present several characterizations of acute perturbations based on the results from [6,31]. In Section 3 we present our new results on the spectral radius and the spectral norm for the difference of BB g − AA D via two examples.
2. Geometrical conditions on the acute perturbation. Let A ∈ C n×n be singular with ind(A) = k > 1 and rank(A k ) = r. Then A has the Jordan canonical form [1,4], for the invertible transformation matrix P such that D ∈ C r×r is nonsingular, N ∈ C (n−r)×(n−r) is nilpotent.
Let C = P −1 AP ∈ C n×n and A π = I n − AA D . It follows from [4, Theorem 7.2.1] that A D and A π are given by Let P = (P 1 P 2 ), where P 1 and Q * 1 have the same column dimensions as D. It is obvious that [27,28] AA D = P 1 Q 1 , I n − AA D = P 2 Q 2 , Q 1 P 1 = I r , Motivated by the acute perturbation of the Moore-Penrose inverse [21,26], weighted Moore-Penrose inverse [17] and the group inverse [31], we present the acute perturbation for the Drazin inverse with respect to the spectral radius instead of the spectral norm, which extends the recent results on the group inverse by Wei in [31].
We present the geometrical conditions for the stable perturbation for the Drazin inverse.

Lemma 2.3 was investigated in
Then A is group invertible if and only if I r + T S is nonsingular. In this case, A g and A π = I n − AA g can be presented by and Now we study the expression B g and BB g for the perturbation of the Drazin inverse.
Then ind(B) = 1 with the group inverse, and the spectral projection, Proof. Let the perturbation matrix be E and If rank(B) = rank(A k ), then it follows from [15,29] that AA D ≤ 1/2, then both I n + EA D and I n + A D E are nonsingular and It is obvious that D + F 11 = Q 1 (A + E)P 1 and D −1 = Q 1 A D P 1 with since Q 1 P 1 = I r and I n − AA D = AA D (see [24] or [28, Lemma 2.3]) and consequently, It is easy to verify that Thus and by simple computations, we obtain We can present the estimation for spectral radius of BB g −AA D , which sharpens the upper bound of BB g − AA g [31, Theorem 2.1 (a)].
Theorem 2.5. Let B = A + E ∈ C n×n with ind(A) = k and rank(A k ) = rank(B).

If the perturbation E satisfies
Then ind(B)=1 and the spectral radius of BB g (I n − AA D ) and AA D (I n − BB g ) are exactly the same, such that Proof. It follows from Lemma 2.3 that and P −1 AA D (I n − BB g )P = P −1 AA D P P −1 (I n − BB g )P We obtain From the proof of Lemma 2.3, it is obvious that Since E is small and (I n + Y ) −1 AA D = AA D (I n + Y ) −1 , then the spectral radius of I r − X −1 satisfies .
and we can estimate the spectral radius of BB g (I n − AA D ).
Next we reveal the relationship between ρ(BB g − AA D ) and ρ[BB g (I n − AA D )]. It follows from Lemma 2.3 that It is obvious that the spectral radius of F 21 (D + F 11 ) −1 X −1 (D + F 11 ) −1 F 12 are the same as  Proof. Denote H = BB g −AA D . For a sufficiently small > 0, there exists a matrix norm [11] · * so that H * ≤ ρ(H) + < 1. It is obvious that I n − (BB g − AA D ) is invertible.
Based on the well known result [10, Theorem 2.6.4]: if H * < 1 for a matrix norm, then the spectral projection matrices BB g and AA D have the same rank, i.e., rank(B) = rank(BB g ) = rank(AA D ) = rank(A k ).
Now we present a necessary and sufficient condition for the acute perturbation of the Drazin inverse, which coincides with the stable perturbation of the Drazin inverse [6]. With the help of Theorem 2.5, we can prove that ρ(BB g − AA D ) < 1. (2) I n − (BB g − AA D ) 2 is invertible. If B is not acute perturbation of A and rank(B) ≥ rank(A k ), then BB g − AA D ≥ 1 (see [28,35]).

Examples.
In this section, we provide two examples to illustrate the difference between the acute perturbation for the Drazin inverse and the spectral norm.
so B is an acute perturbation of A, and In this case, ρ(BB g − AA D ) is very close to BB g − AA D .
4. Concluding remarks. For the general case of A ∈ C n×n with ind(A) = k ≥ 1, and B is an acute perturbation of A with ind(B) = j ≥ 1, Wei [31] conjectured that which is equivalent to the rank condition [5], rank(B j ) = rank(A k ) = rank(A k B j A k ).
We only prove the correctness of the conjecture in a special case of ind(B) = 1 in this paper.
It will be our future research work for the general case of ind(B) > 1, which will be reported in a forthcoming paper.
Very recently, Ji and Wei [12] extend the notion of the Drazin inverse of a square matrix to an even-order square tensor with Einstein product. It will be very interesting to investigate the perturbation bounds for the Drazin inverse in the tensor case.