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Displacement structure of the DMP inverse

  • Jin Zhong EMAIL logo and Hong Yang
Published/Copyright: October 13, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

A matrix A is said to have the displacement structure if the rank of the Sylvester displacement A U − V A or the Stein displacement A − V A U is much smaller than the rank of A . In this article, we study the displacement structure of the DMP inverse A d , † . Estimations for the Sylvester displacement rank of DMP inverse are presented under some restrictions. The generalized displacement is also discussed. The general results are applied to the core inverse.

Keywords: DMP inverse; displacement rank; estimation
MSC 2010: 15A09; 15A10

1 Introduction

Let C m × n be the set of all m × n complex matrices. For A ∈ C m × n , ℛ ( A ) , Ker ( A ) , A ∗ , and A T denote the range, kernel, conjugate transpose, and transpose of A , respectively. I n is the identity matrix of order n . The index of A ∈ C n × n , denoted by Ind ( A ) , is the smallest nonnegative integer k such that rank ( A k ) = rank ( A k + 1 ) . The present article is to study the displacement structure of the DMP inverse [1], which is motivated by [2,3,4, 5,6,7, 8,9,10]. To begin with, we shall recall definitions of some generalized inverses.

The Moore-Penrose inverse of a matrix A ∈ C m × n , denoted by A † , is the unique matrix X ∈ C n × m satisfying the following Penrose equations [11]:

A X A = A , X A X = X , ( A X ) ∗ = A X , ( X A ) ∗ = X A .

The Drazin inverse of A ∈ C n × n is the unique matrix X = A d such that

A k + 1 X = A k , X A X = X , A X = X A ,

where k is the index of A . When Ind ( A ) = 1 , A d is called the group inverse of A and is denoted by A # , and see [11].

In [12], Baksalary and Trenkler introduced the notion of core inverse for a square matrix of index 1. Let A ∈ C n × n be with Ind ( A ) = 1 . Then the core inverse of A is defined as the unique matrix X ∈ C n × n satisfying

A X = A A † , ℛ ( X ) ⊆ ℛ ( A )

and is denoted by X = A # ○ . It is known that A # ○ = A # A A † .

Then three generalizations of the core inverse were recently introduced for n × n complex matrices, namely, core-EP inverse [13], BT inverse [14], and DMP inverse [15]. The DMP inverse [16] of A ∈ C n × n , denoted as A d , † , is the unique solution to the system of equations:

X A X = X , X A = A d A , A k X = A k A † ,

where k = Ind ( A ) . It was shown that A d , † = A d A A † . Thus, if Ind ( A ) = 1 , then A d , † = A # A A † = A # ○ , the core inverse of A . For more properties of the DMP inverse, we refer to [17,18, 19,20].

Furthermore, the definition of the DMP inverse of a square matrix was extended to rectangular matrices in [21], which is called the W -weighted DMP inverse.

Malik and Thome [15] gave the canonical form for the DMP inverse of a square matrix by using the Hartwig-Spindelböck decomposition (see [22]), which is useful in analyzing the properties of the DMP inverse. For any A ∈ C n × n of rank r > 0 , the Hartwig-Spindelböck decomposition is given by

(1) A = S Σ K Σ L 0 0 S ∗ ,

where S ∈ C n × n is unitary, Σ = diag ( σ 1 I r 1 , σ 2 I r 2 , … , σ t I r t ) is a diagonal matrix, and the diagonal entries σ i ( 1 ≤ i ≤ t ) are the singular values of A with σ 1 > σ 2 > ⋯ > σ t > 0 , r 1 + r 2 + ⋯ + r t = r and K ∈ C r × r , L ∈ C r × ( n − r ) satisfy K K ∗ + L L ∗ = I r .

Lemma 1.1

[15] Let A ∈ C n × n be of the form (1). Then

(2) A d , † = S ( Σ K ) d 0 0 0 S ∗ .

Lemma 1.2

[15] Let A ∈ C n × n be a matrix of index m written as (1). Then Ind ( Σ K ) = m − 1 .

The concept of the displacement structure was presented in [23] for the inverse of an integral operator with a convolution kernel. A matrix A is said to have the displacement structure if we can find two-dimensionally compatible matrices U and V such that the rank of the Sylvester displacement A U − V A or the Stein displacement A − V A U is much smaller than the rank of A [24]. It is well known that fast inversion algorithms for a matrix A can be constructed if A is a matrix with a displacement structure. The displacement structure is commonly exploited in the computation of generalized inverses. In recent years, displacement structures of various generalized inverses, such as Moore-Penrose inverse, weighted Moore-Penrose inverse, group inverse, M -group inverse, Drazin inverse, W -weighted Drazin inverse, and core inverse, were studied, and see [2,3,4, 5,6,8, 9,10].

In this article, we will give an upper bound for the Sylvester displacement rank of the DMP inverse under some restrictions in Section 2. In Section 3, we will consider a more general displacement. An estimation for the generalized displacement rank of the DMP inverse under some restrictions is presented. The general results are applied to the core inverse.

2 Sylvester displacement rank

Let A ∈ C m × n and let U ∈ C n × n , V ∈ C m × m be some fixed matrices. The operator

d ( U , V ) A = A U − V A

is called the Sylvester displacement of A . The rank of d ( U , V ) A is called the Sylvester displacement rank of A . It is well known that the Sylvester displacement rank of a Toeplitz rank is at most 2. This low displacement rank property can be exploited to develop fast algorithms for triangular factorization, and inversion, among others [24]. For a nonsingular matrix A , the equality A U − V A = A ( U A − 1 − A − 1 V ) A tells us that the Sylvester displacement rank of a nonsingular matrix A equals that of its inverse A − 1 . In other words, if A is structured with respect to the Sylvester displacement rank associated with ( U , V ) , then its inverse A − 1 is also structured with respect to the Sylvester displacement rank associated with ( V , U ) . It is natural to consider the displacement rank of generalized inverses. In this section, we will give an upper bound for the displacement rank of DMP inverse under some restrictions.

Lemma 2.1

[11] Let A , P ∈ C n × n satisfy P 2 = P . Then

  1. P A = A if and only if ℛ ( A ) ⊂ ℛ ( P ) .

  2. A P = A if and only if Ker ( P ) ⊂ Ker ( A ) .

Define the matrices

M = A d , † A , N = I − M , M ∗ = A A d , † , N ∗ = I − M ∗ .

Then it is easy to see that A d , † ( A U − V A ) A d , † = ( I − N ) U A d , † − A d , † V ( I − N ∗ ) . Thus, we obtain the following lemma.

Lemma 2.2

Let A , U , V ∈ C n × n . Then

(3) A d , † V − U A d , † = A d , † V N ∗ − N U A d , † − A d , † ( A U − V A ) A d , † .

By considering the ranks of both sides in equality (3), we can obtain an upper bound for the Sylvester displacement rank of A d , † .

Theorem 2.1

The V U -displacement rank of A d , † satisfies the following estimate:

(4) rank ( A d , † V − U A d , † ) ≤ rank ( A U − V A ) + rank ( M ∗ V N ∗ ) + rank ( N U M ) .

Proof

It is clear that ℛ ( M ) = ℛ ( A d , † A ) ⊂ ℛ ( A d , † ) . Since

ℛ ( A d , † ) = ℛ ( A d A A † ) ⊂ ℛ ( A d A ) = ℛ ( A d , † A ) = ℛ ( M ) ,

then ℛ ( M ) = ℛ ( A d , † ) .

On the other hand, it is clear that Ker ( A d , † ) ⊂ Ker ( A A d , † ) = Ker ( M ∗ ) . Conversely, for any x ∈ Ker ( M ∗ ) , A A d , † x = A A d A A † x = 0 , which implies that A d A A d A A † x = A d A A † x = A d , † x = 0 , i.e., Ker ( M ∗ ) ⊂ Ker ( A d , † ) . Therefore, Ker ( A d , † ) = Ker ( M ∗ ) .

It follows that

rank ( N U A d , † ) = dim [ ℛ ( N U A d , † ) ] = dim [ N U ℛ ( A d , † ) ] = dim [ N U ℛ ( M ) ] = dim [ ℛ ( N U M ) ] = rank ( N U M )

and

rank ( A d , † V N ∗ ) = n − dim [ Ker ( A d , † V N ∗ ) ] = n − dim [ Ker ( M ∗ V N ∗ ) ] = rank ( M ∗ V N ∗ ) .

Now, the inequality (4) follows immediately from Lemma 2.2.□

Next we give an upper bound for the sum of the second and third terms on the right-hand side of (4) under some restrictions.

Theorem 2.2

Let A ∈ C n × n be of the form (1) and let

U = S U 11 U 12 U 21 U 22 S ∗ ∈ C n × n , V = S V 11 V 12 V 21 V 22 S ∗ ∈ C n × n ,

where U 11 , V 11 ∈ C r × r . If ℛ ( U 11 ) ⊂ ℛ [ ( Σ K ) p − 1 ] and Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V 11 ) , then

(5) rank ( M ∗ V N ∗ ) + rank ( N U M ) ≤ rank ( U G − G V ) ,

where G = A p A † , p = Ind ( A ) .

Proof

We first determine the structure of G . If A has the form (1), then it is not difficult to see that

A † = S K ∗ Σ − 1 0 L ∗ Σ − 1 0 S ∗ .

Thus,

G = A p A † = S ( Σ K ) p ( Σ K ) p − 1 Σ L 0 0 S ∗ S K ∗ Σ − 1 0 L ∗ Σ − 1 0 S ∗ = S ( Σ K ) p K ∗ Σ − 1 + ( Σ K ) p − 1 Σ L L ∗ Σ − 1 0 0 0 S ∗ = S ( Σ K ) p − 1 Σ ( K K ∗ + L L ∗ ) Σ − 1 0 0 0 S ∗ = S ( Σ K ) p − 1 0 0 0 S ∗ .

Next, we determine the block representations of M ∗ V N ∗ and N U M . Denoted P = ( Σ K ) ( Σ K ) d . Then

(6) M ∗ V N ∗ = A A d , † V ( I − A A d , † ) = S P 0 0 0 V 11 V 12 V 21 V 22 I − P 0 0 I S ∗ = S P V 11 ( I − P ) P V 12 0 0 S ∗ .

Since Ind ( A ) = p , then by Lemma 1.2, Ind ( Σ K ) = p − 1 . If Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V 11 ) , then Ker ( P ) = Ker [ ( Σ K ) d ] = Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V 11 ) . Thus, by Lemma 2.1, V 11 ( I − P ) = 0 . Now, it can be seen from (6) that

(7) rank ( M ∗ V N ∗ ) = rank ( P V 12 ) = n − dim Ker ( P V 12 ) = n − dim Ker [ ( Σ K ) p − 1 V 12 ] = rank [ ( Σ K ) p − 1 V 12 ] .

On the other hand,

(8) N U M = ( I − A d , † A ) U ( A d , † A ) = S I − P − Q 0 I U 11 U 12 U 21 U 22 P Q 0 0 S ∗ = S ( I − P ) U 11 P − Q U 21 P ( I − P ) U 11 Q − Q U 21 Q U 21 P U 21 Q S ∗ ,

where Q = ( Σ K ) d ( Σ L ) .

We can see from (8) that

(9) rank ( N U M ) = rank ( S I Q 0 I S ∗ N U M ) = rank ( S ( I − P ) U 11 P ( I − P ) U 11 Q U 21 P U 21 Q S ∗ ) = rank ( S ( I − P ) U 11 0 U 21 P 0 P Q 0 0 S ∗ ) ≤ rank ( ( I − P ) U 11 0 U 21 P 0 ) .

Notice that ℛ [ ( Σ K ) p − 1 ] = ℛ [ ( Σ K ) d ] = ℛ ( P ) . If ℛ ( U 11 ) ⊂ ℛ [ ( Σ K ) p − 1 ] , then by Lemma 2.1, P U 11 = U 11 . It follows from (9) that

(10) rank ( N U M ) ≤ rank ( U 21 P ) = rank [ U 21 ( Σ K ) d ] = rank [ U 21 ( Σ K ) p − 1 ] .

Set F = U G − G V . Then

S ∗ F S = S ∗ U S S ∗ G S − S ∗ G S S ∗ V S = U 11 U 12 U 21 U 22 ( Σ K ) p − 1 0 0 0 − ( Σ K ) p − 1 0 0 0 V 11 V 12 V 21 V 22 = U 11 ( Σ K ) p − 1 − ( Σ K ) p − 1 V 11 − ( Σ K ) p − 1 V 12 U 21 ( Σ K ) p − 1 0 .

It follows from [25], (7), and (10) that

rank ( U 11 ( Σ K ) p − 1 − ( Σ K ) p − 1 V 11 − ( Σ K ) p − 1 V 12 U 21 ( Σ K ) p − 1 0 ) ≥ rank [ ( Σ K ) p − 1 V 12 ] + rank [ U 21 ( Σ K ) p − 1 ] ≥ rank ( M ∗ V N ∗ ) + rank ( N U M ) ,

which completes the proof.□

We remark that if the two conditions ℛ ( U 11 ) ⊂ ℛ [ ( Σ K ) p − 1 ] and Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V 11 ) were removed from Theorem 2.2, then the inequality (5) may not hold. Let’s give an example to show this.

Example 2.1

Let

A = 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 .

Then rank ( A ) = 4 and rank ( A 2 ) = rank ( A 3 ) = 3 . Thus, Ind ( A ) = 2 . The singular value decomposition of A is

A = S Σ ˜ Q T = 0 − 1 2 0 0 1 2 − y 0 0 x 0 0 0 1 0 0 0 − 1 2 0 0 − 1 2 − x 0 0 − y 0 5 + 1 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 5 − 1 2 0 0 0 0 0 0 − x 0 0 y 0 0 0 1 2 0 1 2 0 − 1 0 0 0 − y 0 0 − x 0 0 0 1 2 0 − 1 2 = S Σ 0 0 0 Q T S S T = S Σ 0 0 0 K L M N S T = S Σ K Σ L 0 0 S T ,

where x = 5 − 5 10 , y = 5 + 5 10 , and Σ K = 0 ( x + y ) ( 5 + 1 ) 2 2 0 0 0 0 − 2 0 − ( x + y ) 0 0 x − y 0 ( x − y ) ( 5 − 1 ) 2 2 0 0 .

Let

U = V = S I 4 0 0 0 S T = 1 2 0 0 1 2 0 0 1 0 0 0 0 0 1 0 0 1 2 0 0 1 2 0 0 0 0 0 1 .

Then it can be seen from rank ( Σ K ) = 3 and rank ( U 11 ) = rank ( V 11 ) = 4 that ℛ ( U 11 ) ⊄ ℛ [ ( Σ K ) ] and Ker [ ( Σ K ) ] ⊄ Ker ( V 11 ) .

A direct calculation shows that

M = A d , † A = 1 3 0 0 2 3 0 0 2 3 0 0 2 3 0 0 1 0 0 1 3 0 0 2 3 0 0 1 3 0 0 1 3 , M ∗ = A A d , † = 1 2 0 0 1 2 0 0 2 3 0 0 2 3 0 0 1 0 0 1 2 0 0 1 2 0 0 1 3 0 0 1 3

and

G = A 2 A † = 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 2 0 0 1 2 0 .

Now, it can be seen that

rank ( M ∗ V N ∗ ) = 3 , rank ( N U M ) = 2 , rank ( U G − G V ) = 2 ,

i.e., rank ( M ∗ V N ∗ ) + rank ( N U M ) > rank ( U G − G V ) .

By combining Theorems 2.1 and 2.2 we have the following.

Corollary 2.1

Let A ∈ C n × n be of the form (1) and let

U = S U 11 U 12 U 21 U 22 S ∗ ∈ C n × n , V = S V 11 V 12 V 21 V 22 S ∗ ∈ C n × n ,

where U 11 , V 11 ∈ C r × r . If ℛ ( U 11 ) ⊂ ℛ [ ( Σ K ) p − 1 ] and Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V 11 ) , then

rank ( A d , † V − U A d , † ) ≤ rank ( A U − V A ) + rank ( U G − G V ) ,

where G = A p A † , p = Ind ( A ) .

Let A ∈ C n × n be of the form (1). If Ind ( A ) = 1 , it was shown in [22] that K is nonsingular. In this case, Σ K is also nonsingular. Then ( Σ K ) p − 1 = I and the two restrictions ℛ ( U 11 ) ⊂ ℛ [ ( Σ K ) p − 1 ] and Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V 11 ) can be removed from Theorem 2.2. Hence, we can obtain an estimate for the V U -displacement rank of the core inverse.

Corollary 2.2

For any A ∈ C n × n with Ind ( A ) = 1 , we have

(11) rank ( A # ○ V − U A # ○ ) ≤ rank ( A U − V A ) + rank ( U G − G V ) ,

where G = A A † .

We remark that the author in [6] showed that (11) holds for G = A A ∗ , while we derive a different G such that the inequality (11) holds.

3 Generalized displacement rank

In this section, we will consider a more general displacement, and the results of Sylvester displacement in Section 2 will be extended.

Let a = [ a i j ] 0 1 be a nonsingular 2 × 2 matrix. For any fixed U ∈ C n × n and V ∈ C m × m , the generalized a ( V , U ) displacement of A ∈ C m × n is defined by

(12) a ( V , U ) A = ∑ i , j = 0 1 a i j V i A U j .

If we set

a = 0 1 − 1 0 and a = 1 0 0 − 1 ,

respectively, in (12), then we obtain the Sylvester displacement and the Stein displacement.

For Z ∈ C n × n and a = [ a i j ] 0 1 ∈ C 2 × 2 , if a 00 I + a 01 Z is nonsingular, then we denote by f a ( Z ) the matrix

f a ( Z ) = ( a 00 I + a 01 Z ) − 1 ( a 10 I + a 11 Z ) .

We first give two lemmas, which are taken from [4].

Lemma 3.1

Let a be a 2 × 2 nonsingular matrix and U ∈ C m × m , V ∈ C n × n . Then there exist 2 × 2 matrices b = [ b i j ] 0 1 and c = [ c i j ] 0 1 such that b 00 I + b 01 V and c 00 I + c 01 U are nonsingular and

a = b T d c for d = 0 1 − 1 0 .

Lemma 3.2

Let b and c be matrices satisfying the conditions in Lemma 3.1. Then for A ∈ C m × n ,

a ( V , U ) A = ( b 00 I + b 01 V ) [ A f c ( U ) − f b ( V ) A ] ( c 00 I + c 01 U ) .

The following results play important roles in obtaining an extension of Corollary 2.1 for generalized a ( U , V ) displacement.

Theorem 3.1

  1. If [ ψ i j ] 0 1 is nonsingular and ψ 00 I + ψ 01 V is nonsingular, then

    rank ( M ∗ V N ∗ ) = rank ( M ∗ V ˜ N ∗ ) ,

    where V ˜ = f ψ ( V ) .

  2. If [ ϕ i j ] 0 1 is nonsingular and ϕ 00 I + ϕ 01 U is nonsingular, then

    rank ( N U M ) = rank ( N U ˜ M ) ,

    where U ˜ = f ϕ ( U ) .

Proof

Define

Ψ = Ker ( G ) ∩ Ker ( G V ) , Ψ 1 = Ker ( G ) ⊝ Ψ ,

where G = A p A † , p = Ind ( A ) . We conclude that Ker ( A d , † ) = Ker ( G ) . Indeed, for any x ∈ Ker ( A d , † ) , A d , † x = A d A A † x = 0 ; thus, A d A d A A † x = A d A † x = 0 . It follows that A † x ∈ K e r ( A d ) = Ker ( A p ) . Hence, A p A † x = 0 , i.e., Ker ( A d , † ) ⊂ Ker ( G ) . Conversely, for any x ∈ Ker ( A p A † ) , we have A † x ∈ Ker ( A p ) = Ker ( A d ) . Then A d A † x = 0 , which gives that A A d A † x = A d A A † x = 0 , i.e., x ∈ Ker ( A d A A † ) = Ker ( A d , † ) . Therefore, Ker ( A d , † ) = Ker ( G ) .

Next, we will show that M ∗ V N ∗ is one-to-one on Ψ 1 . For any x ∈ Ker ( M ∗ V N ∗ ) and x ∈ Ψ 1 , then V N ∗ x ∈ Ker ( M ∗ ) . Since Ker ( M ∗ ) = Ker ( A d , † ) = Ker ( G ) , then G V N ∗ x = G V x = 0 . Now, it is easy to see that x ∈ Ψ ∩ Ψ 1 = { 0 } , i.e., x = 0 .

Furthermore, for any x ∈ Ψ , we have A d , † x = 0 and A d , † V x = 0 . Thus, M ∗ V N ∗ x = M ∗ V ( I − A A d , † ) x = M ∗ V x = A A d , † V x = 0 , i.e., ( M ∗ V N ∗ ) ∣ Ψ = 0 . Now we conclude that rank ( M ∗ V N ∗ ) = dim ( Ψ 1 ) .

Similarly, we define

Ψ ˜ = Ker ( G ) ∩ Ker ( G V ˜ ) , Ψ ˜ 1 = Ker ( G ) ⊝ Ψ ˜ ,

then it follows that rank ( M ∗ V ˜ N ∗ ) = dim ( Ψ ˜ 1 ) .

Now we show that the nonsingular matrix ψ 00 I + ψ 01 V bijectively maps Ψ onto Ψ ˜ . Suppose that x ∈ Ψ , then x , V x ∈ Ker ( G ) . Hence, both y ≡ ( ψ 10 I + ψ 11 V ) x and z ≡ ( ψ 00 I + ψ 01 V ) x are contained in Ker ( G ) . Since

y = ( ψ 10 I + ψ 11 V ) x = ( ψ 00 I + ψ 01 V ) − 1 ( ψ 00 I + ψ 01 V ) ( ψ 10 I + ψ 11 V ) x = ( ψ 00 I + ψ 01 V ) − 1 ( ψ 10 I + ψ 11 V ) ( ψ 00 I + ψ 01 V ) x = V ˜ z ,

then z , V ˜ z ∈ Ker ( G ) , which implies that z ∈ Ψ ˜ . Conversely, in a similar way, we can obtain that ( ψ 00 I + ψ 01 V ) − 1 z ∈ Ψ for z ∈ Ψ ˜ .

Therefore,

rank ( M ∗ V N ∗ ) = dim ( Ψ 1 ) = dim [ Ker ( G ) ] − dim ( Ψ ) = dim [ Ker ( G ) ] − dim ( Ψ ˜ ) = dim ( Ψ ˜ 1 ) = rank ( M ∗ V ˜ N ∗ ) ,

which proves the assertion ( a ) .

Assertion ( b ) can be proved analogously.□

Now we can generalize Corollary 2.1 for general a ( U , V ) displacement.

Theorem 3.2

Let a and b be 2 × 2 nonsingular matrices and let A ∈ C n × n be of the form (1). Suppose that

f w ( U ) = S U ˜ 11 U ˜ 12 U ˜ 21 U ˜ 22 S ∗ , f z ( V ) = S V ˜ 11 V ˜ 12 V ˜ 21 V ˜ 22 S ∗ .

If ℛ ( U ˜ 11 ) ⊂ ℛ [ ( Σ K ) p − 1 ] and Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V ˜ 11 ) , Then

rank [ a ( U , V ) A d , † ] ≤ rank [ a T ( V , U ) A ] + rank [ b ( U , V ) G ] ,

where G = A p A † and p = Ind ( A ) .

Proof

By Lemma 3.1, there exist 2 × 2 matrices w , x , y , z such that w 00 I + w 01 U , x 00 I + x 01 V , y 00 I + y 01 U , and z 00 I + z 01 V are nonsingular and a = w T d z , b = x T d y .

By Lemma 3.2,

rank [ a ( U , V ) A d , † ] = rank [ A d , † f z ( V ) − f w ( U ) A d , † ]

and

rank [ a T ( V , U ) A ] = rank [ A f w ( U ) − f z ( V ) A ] .

Moreover, it follows from Theorem 2.1 that

rank [ A d , † f z ( V ) − f w ( U ) A d , † ] ≤ rank [ A f w ( U ) − f z ( V ) A ] + rank [ M ∗ f z ( V ) N ∗ ] + rank ( N f w ( U ) M ] .

Hence, according to Corollary 2.1 and Theorems 2.2 and 3.1, we obtain

□ rank [ a ( U , V ) A d , † ] − rank [ a T ( V , U ) A ] = rank [ f w ( U ) A d , † − A d , † f z ( V ) ] − rank [ A f w ( U ) − f z ( V ) A ] ≤ rank [ N f w ( U ) M ] + rank [ M ∗ f z ( V ) N ∗ ] = rank [ N f y ( U ) M ] + rank [ M ∗ f x ( V ) N ∗ ] ≤ rank [ f y ( U ) G − G f x ( V ) ] = rank [ b ( U , V ) G ] .

Corollary 3.1

Let a and b be 2 × 2 nonsingular matrices. Then

(13) rank [ a ( U , V ) A # ○ ] ≤ rank [ a T ( V , U ) A ] + rank [ b ( U , V ) G ] ,

where G = A A † .

Example 3.1

Denote

U = 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 , V = 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0

as the shift-down matrix and the shift-up matrix, respectively. Let

A = 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0

be a singular Toeplitz matrix with Ind ( A ) = 1 and rank ( A ) = 5 .

The core inverse of A is

A # ○ = − 1 20 − 11 20 9 20 1 4 3 10 − 9 20 7 20 − 3 20 − 3 20 1 4 − 1 10 3 20 − 1 5 4 5 − 1 5 0 1 5 1 5 − 3 20 7 20 7 20 − 1 4 − 1 10 13 20 − 7 20 3 20 3 20 3 4 1 10 − 3 20 9 20 − 1 20 − 1 20 − 1 4 3 10 1 20 .

If we choose

a = b = 1 0 0 − 1

in Corollary 3.1, then a direct computation shows that

rank [ a ( U , V ) A # ○ ] = 4 , rank [ a T ( V , U ) A ] = 2 , rank [ b ( U , V ) A A † ] = 5 ,

i.e., the estimate (13) in Corollary 3.1 holds.

4 Computation of the displacement

In this section, we study the explicit form of the displacement, and we give an expression for the displacement of the DMP inverse of A through DMP inverse solutions of some special linear systems of equations. For simplicity, we only consider the Sylvester displacement, and the starting point is (3).

(1) To compute A d , † ( A U − V A ) A d , † , first find a full-rank decomposition of A U − V A ,

A U − V A = G F ∗ = ∑ i = 1 r g i f i ∗ ,

where, G = [ g 1 , g 2 , … , g r ] ∈ C n × r and F = [ f 1 , f 2 , … , f r ] ∈ C n × r are of rank r , and then compute the DMP inverse solutions A d , † g i and f i ∗ A d , † .

(2) Next we show how to compute A d , † V ( I − A A d , † ) . We start with a full-rank decomposition

A d , † V ( I − A A d , † ) = M N ∗ .

Now we determine the kernels of A p A † ( p = Ind ( A ) ) and the matrix

C = A p A † N ∗

and a set of vectors w 1 , w 2 , … , w p forming an orthogonal basis for the orthogonal complements of Ker ( C ) in Ker ( A p A † ) . We generate the matrix W = [ w 1 , w 2 , … , w p ] . Then we have

Ker ( A p A † ) = Ker ( C ) ⊕ ℛ ( W ) .

The matrix R = W W ∗ is an orthogonal projection onto ℛ ( W ) . We have

R ( I − A A d , † ) = W W ∗ ( I − A A d , † ) = W W ∗ = R

and

A d , † V ( I − R ) ( I − A A d , † ) = 0 .

Hence,

A d , † V ( I − A A d , † ) = A d , † V R = A d , † V W W ∗ .

(3) We proceed analogously for ( I − A d , † A ) U A d , † . We obtain

( I − A d , † A ) U A d , † = S U A d , † = Z Z ∗ U A d , † ,

where S is the orthogonal complement onto Ker ( A ) ⊖ Ker ( C ∗ ) with C ∗ defined by C ∗ ∗ = [ A p A † , X ∗ ] and Z being a matrix with columns forming an orthogonal basis of ℛ ( S ) .

To compute A d , † V − U A d , † , one has to find 2 r DMP inverse solutions A d , † g i and f i ∗ A d , † , where r = rank ( A U − V A ) , and the p + q DMP inverse solutions A d , † V w i ( 1 ≤ i ≤ p ) and z j ∗ U A d , † ( 1 ≤ j ≤ q ) , where p + q ≤ rank ( A p A † V − U A p A † ) .

5 Concluding remark

In this article, we studied the displacement structure of the DMP inverse. An upper bound for the Sylvester displacement rank of the DMP inverse was given under some restrictions. An example was presented to show that these restrictions can not be removed. Estimations for the general displacement rank of the DMP inverse were also investigated. As corollaries, estimations for the Sylvester displacement rank and the general displacement rank of the core inverse without any restrictions were obtained.

Furthermore, the theorems obtained before can be applied to many classical structured matrices, such as Toeplitz matrices, Hankel matrices, and Cauchy matrices. For simplicity, we only consider close-to-Toeplitz matrices.

Close-to-Toeplitz matrices are a class of matrices whose U V -displacement ranks are small compared with the sizes of U and V being (forward or backward) (block) shifts, including Toeplitz matrices, Hankel matrices, and more general block matrices with Toeplitz or Hankel blocks, and sums, products, and inverses of these matrices.

Let U be the shift-down matrix

Z = 0 1 0 ⋱ ⋱ 1 0 ,

V be the shift-up matrix Z ∗ .

If ℛ ( U ˜ 11 ) ⊂ ℛ [ ( Σ K ) p − 1 ] and Ker [ ( Σ K ) p − 1 ] ⊂ Ker ( V ˜ 11 ) , then choosing

a = b = 1 0 0 − 1

in Theorem 3.2 gives

(14) rank ( A d , † − Z A d , † Z ∗ ) ≤ r + + r A p A † ,

where r + = rank ( A − Z ∗ A Z ) , r A p A † = rank ( A p A † − Z A p A † Z ∗ ) , and p = Ind ( A ) .

If the estimate of (14) is small for a close-to-Toeplitz matrix, then it leads to the famous representation Gohberg-Semencul type of A d , † :

A d , † = ∑ k = 1 r L k U k ,

where r is the displacement rank of A d , † , i.e., r + + r A p A † in (14).

As a corollary of (14), for core inverse, we have

rank ( A # ○ − Z A # ○ Z ∗ ) ≤ r + + r A A † ,

where r + = rank ( A − Z ∗ A Z ) and r A A † = rank ( A A † − Z A A † Z ∗ ) .

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions.

  1. Funding information: This research was supported by the National Natural Science Foundation of China (Grant No. 12261043), the Program of Qingjiang Excellent Young Talents, and Jiangxi University of Science and Technology (JXUSTQJYX2017007).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no dataset were generated or analysed during the current study.

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Received: 2021-07-07
Revised: 2022-08-30
Accepted: 2022-09-13
Published Online: 2022-10-13

© 2022 Jin Zhong and Hong Yang, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0506
Keywords for this article
DMP inverse; displacement rank; estimation
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  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
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  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
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  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
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  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
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