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The uniqueness of expression for generalized quadratic matrices

  • Meixiang Chen EMAIL logo , Zhongpeng Yang and Qinghua Chen
Published/Copyright: March 20, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

It is shown that the expression as A 2 = α A + β P for generalized quadratic matrices is not unique by numerical examples. Then it is proven that the uniqueness of expression for generalized quadratic matrices is concerned not only with the properties of A but also with the rank of P . Furthermore, the sufficient and necessary conditions for the uniqueness of generalized quadratic matrices’expression are obtained. Finally, some related discussions about generalized quadratic matrices are also given.

Keywords: generalized quadratic matrix; uniqueness; linear combination; rank
MSC 2010: 15A09; 15A18

1 Introduction

Let C n × n and C [ x ] be the sets of all n × n matrices and polynomials over the complex number field C , respectively, C ∗ = C \ { 0 } be the set of nonzero complex numbers, Z + be the set of all the positive integer numbers, and i ∈ C be the square root of − 1 , namely, i 2 = − 1 . And ∣ α ∣ stands for the modulus of α ∈ C . The symbols r ( A ) and t r ( A ) denote the rank and trace of a matrix A , and I n denotes the n × n identity matrix, sometimes simply write I if the size is immaterial.

In 2005, Farebrother and Trenkler [1] extended the concept of quadratic matrix and defined the set Ω n ( P ) as follows, for a given idempotent matrix P ,

(1.1) Ω n ( P ) = { A ∈ C n × n ∣ A 2 = α A + β P , A P = P A = A , for some α , β ∈ C } .

By [1], A ∈ Ω n ( P ) is called as a generalized quadratic matrix with respect to P . Clearly 0 and P ∈ Ω n ( P ) is trivial. From now on, unless otherwise specified, both A ∈ Ω n ( P ) and P are nonzero.

When P = I n , a subclass of Ω n ( P ) was investigated by Aleksiejczyk and Smoktynowicz [2], where A satisfies the quadratic equation ( A − d I n ) ( A − e I n ) = 0 , with d , e ∈ C .

Moreover, by (1.1), Ω n ( P ) contains the following subclasses (see [1, Introduction], P ≠ I n ):

idempotent matrices A 2 = A ( α = 1 , β = 0 ) generalized involutory matrices A 2 = P ( α = 0 , β = 1 ) nilpotent matrices A 2 = 0 ( α = 0 , β = 0 ) skew-idempotent matrices A 2 = − A ( α = − 1 , β = 0 ) generalized skew-involutory matrices A 2 = − P ( α = 0 , β = − 1 )

In 2007, generalized quadratic operator in the form of (1.1) was introduced by Duan and Du [3], further researches on generalized quadratic operators were given in [4,5], and the subject had been discussed in depth from different angles in [6–28]. Many common matrix classes are included in the generalized quadratic matrices. Of course, with the development of researches, the discussion is more and more complexity.

Among the known results of generalized quadratic matrices, the basic properties of generalized quadratic matrices shown by Farebrother and Trenkler [1] play important roles.

Proposition 1.1

(see [1, Theorems 1 and 2]) Let A ∈ Ω n ( P ) , that is,

(1.2) A 2 = α A + β P , f o r s o m e c o m p l e x n u m b e r s α a n d β ,

then for all k ∈ Z + , it has

(1.3) A k + 1 = α k A + β k P ∈ Ω n ( P ) ,

where

(1.4) α 1 = α , β 1 = β , α k + 1 = α α k + β k , β k + 1 = β α k .

For any set of complex numbers κ 0 , κ 1 , … , κ m , let the polynomial

(1.5) κ ( x ) = κ m x m + κ m − 1 x m − 1 + ⋯ + κ 1 x + κ 0 ∈ C [ x ] .

Farebrother and Trenkler [1] also show that there is the finite order polynomial

(1.6) κ ( A ) = κ m A m + κ m − 1 A m − 1 + ⋯ + κ 1 A + κ 0 P ∈ Ω n ( P ) , where A ∈ Ω n ( P ) .

Huang et al. [21] discussed the rank of matrices as of the form in (1.6).

Proposition 1.2

(see [1, (2.1–2.3)]) Let A ∈ Ω n ( P ) with A 2 = α A + β P . Then

(1.7) κ ( A ) = α * A + β * P ∈ Ω n ( P ) ,

(1.8) α * = ∑ j = 2 m κ j α j − 1 + κ 1 , β * = ∑ j = 2 m κ j β j − 1 + κ 0 .

Farebrother and Trenkler [1] simplified the system of difference equations from Proposition 1.1 charactering α k and β k to the difference equation:

(1.9) α k + 2 − α k + 1 α − α k β = 0 .

Once α n is determined, we obtain the solution for β n by the identity β n = β α n . The characteristic equation belonging to (1.9) is given by Goldberg [29, Sec. 5.12]

x 2 − α x − β = 0 ,

with solutions x 1 = α 2 + γ and x 2 = α 2 − γ , where γ is one square root of α 2 4 + β . Then

Proposition 1.3

(see [1, P248]) Let A ∈ Ω n ( P ) with A 2 = α A + β P , x 1 = α 2 + γ , x 2 = α 2 − γ , where γ is one square root of α 2 4 + β . Then α k in (1.3) can be expressed as follows:

α k = 1 2 γ ( x 1 k + 1 − x 2 k + 1 ) , i f γ ≠ 0 ; ( 1.10 ) ( k + 1 ) α 2 k , i f γ = 0 . ( 1.11 )

Proposition 1.4

(see [1, Theorem 4]) Let A ∈ Ω n ( P ) such that A 2 = α A + β P and B = ω A + μ P . Then A B A = A if and only if

(1.12) β = 0 a n d ω α 2 + μ α = 1 ; o r β ≠ 0 , ω = 1 β a n d μ = − α β .

In view of studies by Goldberg [29] and Israel and Greville [30], B = ω A + μ P in Proposition 1.4 is a g -inverse of A .

2 Statement of problem

Example 2.1

Let

(2.1) A 1 = 1 3 2 − 12 10 − 2 − 2 0 2 2 − 2 − 6 8 2 − 2 − 6 8 2 , A 2 = 1 3 − 5 − 6 5 − 1 − 1 − 6 1 1 − 1 − 3 − 2 1 − 1 − 3 4 − 5 , P = 1 3 2 6 − 5 1 1 3 − 1 − 1 1 3 − 1 − 1 1 3 − 4 2 .

In view of the study by Uç et al. [14, Example 3], both A 1 and A 2 are quadratic matrices with ( A 1 − 2 I ) A 1 = 0 , ( A 2 + 2 I ) ( A 2 + I ) = 0 , respectively, and P = P 2 . By calculation, we obtain

(2.2) A 1 = 2 I − 2 P , A 2 = − I − P .

From the study by Uç et al. [14, Example 3], if a 1 = − 3 4 and a 2 = − 3 2 , then A = a 1 A 1 + a 2 A 2 satisfies ( A − 3 P ) ( A + P ) = A 2 − 2 A − 3 P = 0 , and according to (1.1),

(2.3) A 2 = 2 A + 3 P , A P = P A = A ∈ Ω 4 ( P ) , α = α 1 = 2 , β = β 1 = 3 .

By applying (2.2) and the study by Uç et al. [14, Example 3], we obtain A = − 3 4 A 1 − 3 2 A 2 = 3 P , so

(2.4) A = 3 P ∈ Ω 4 ( P ) , A 2 = 9 P , A 3 = 27 P .

  1. From (2.4), it follows

    (2.5) 8 A + 3 P = A 3 .

    It differs from α 2 = α α 1 + β 1 = 7 and β 2 = β α 1 = 6 calculating from (1.3) and (1.4).

  2. Let κ ( x ) = x 3 + x 2 + x + 1 ∈ C [ x ] , by (1.5), (1.6), and (2.4), it follows

    (2.6) 12 A + 4 P = κ ( A ) ( = A 3 + A 2 + A + P = 40 P ) ,

    which is different from α * = α 1 + α 2 + 1 = 10 ≠ 12 , β * = β 1 + β 2 + 1 = 10 ≠ 4 by applying (1.7) and (1.8).

  3. From Proposition 1.3 and (2.3), γ = α 2 4 + β = 2 , x 1 = α 2 + γ = 3 , x 2 = α 2 − γ = − 1 . Then combining with (1.4) and (1.10) yields α 2 = 1 2 γ ( x 1 2 + 1 − x 2 2 + 1 ) = 7 , β 2 = β α 1 = 6 , it shows the coefficient for A 3 as a linear combination of A and P couldnot be obtained from (1.4) and (1.10).

  4. Let

    (2.7) B = ω A + μ P , ω = 1 6 , μ = − 1 6 ,

    and combining (2.4) with (2.7) yields that B ∈ Ω n ( P ) and

    (2.8) A B A = ( 3 P ) 1 3 P ( 3 P ) = 3 P = A .

Equation (2.8) indicates that B in (2.7) satisfies the assumption of Proposition 1.4, and by (2.3), we obtain β ( = 3 ) ≠ 0 . However, it follows from (1.12) in Proposition 1.4 that 1 β = 1 3 ≠ 1 6 = ω , − α β = − 2 3 ≠ − 1 6 = μ . Hence, when β ≠ 0 , the coefficients ω , μ for B as the linear combination of A and P are not always satisfied (Proposition 1.4).

A ∈ Ω n ( P ) in Example 2.1 is a numerical example obtained by Uç et al. [14] in 2015. It shows the coefficients for A 2 , A k + 1 , κ ( A ) , and g -inverse of A as linear combinations of A and P couldnot always be determined by the methods given by Propositions 1.1–1.4. Obviously, these conclusions in [1] based on Farebrother and Trenkler [1, Theorem 1] and its algorithm (1.4). For convenience, we list the proof procedure in Farebrother and Trenkler [1, Theorem 1, p. 246] as follows.

By induction on k . The assertion is true for k = 1 . Assume that the following identity is valid:

A k + 1 = α k A + β k P .

We have to show A k + 2 = α k + 1 A + β k + 1 P . This follows from A k + 2 = A A k + 1 = A ( α k A + β k P ) by assumption, and thus,

A k + 2 = α k A 2 + β k A P = α k ( α A + β P ) + β k A = ( α α k + β k ) A + β α k P = α k + 1 A + β k + 1 P .

Seeking into the proof of Farebrother and Trenkler [1, Theorem 1] requires

(2.9) b 1 A + c 1 P = b 2 A + c 2 P ⇔ b 1 = b 2 , c 1 = c 2 , for A ∈ Ω n ( P ) .

However, by observing (2.4) and (2.5) in Example 2.1, for A ∈ Ω 4 ( P ) , it has 8 A + 3 P = 5 A + 12 P ( = A 3 ) , which shows that (2.9) may has a flaw.

Farebrother and Trenkler [1, Introduction] pointed out that “it should be noted that for A ∈ Ω n ( P ) , the expression as A 2 = α A + β P is not unique” and illustrated it by A = I n and P = I n , as A 2 = 1 ⋅ A + 0 ⋅ P = 0 ⋅ A + 1 ⋅ I n . It indicates that Farebrother and Trenkler [1] have noted that (1.1) is not unique, but paid not enough attention to it. From Theorem 1, we see [1] defaulted (2.9) is an identity. Combining with Example 2.1, we should pay attention to the uniqueness of (1.1). When it is unique? When it isn’t? and why? What special properties does the generalized quadratic matrices with (1.1) being not unique have? The answers for these questions are significant for further study on generalized quadratic matrices.

In this article, we pay attention to the uniqueness of expression for generalized quadratic matrices, divide the set Ω n ( P ) into the union of four disjointed subsets, and give a simple and practical method of judging whether the expression is unique, and then illustrate it by examples from previous studies [13,14].

In the following, we make some preparations.

First, similar to [31], we call A ∈ C n × n the scalar-idempotent matrix determined by λ , if there is λ ∈ C ∗ such that A 2 = λ A .

Lemma 2.1

Let A ( ≠ 0 ) ∈ C n × n be a scalar-idempotent matrix determined by λ . Then the scalar λ is unique.

Proof

If A 2 = λ A = λ 0 A , then ( λ − λ 0 ) A = 0 . As A ≠ 0 , then λ = λ 0 . □

Lemma 2.2

(see [11, Lemma 9], [12, Theorem 1.1], [13, Theorem 2.1]) For a given idempotent matrix P ∈ C n × n with r ( P ) = r , let A ∈ C n × n , A P = P A = A . Then there exists an invertible matrix G such that

(2.10) A = G diag ( A 0 , 0 n − r ) G − 1 , P = G diag ( I r , 0 n − r ) G − 1 , A 0 ∈ C r × r .

In view of [32] or [33, Problem 3.4.25] yields the following result.

Lemma 2.3

If the annihilating polynomial of A ∈ C n × n has no multiple roots, then A is similar to a diagonal matrix.

Lemma 2.4

For a given idempotent matrix P ∈ C n × n with r ( P ) = 1 , it has Ω n ( P ) = { λ P ∣ λ ∈ C } .

Proof

Obviously, { λ P ∣ λ ∈ C } ⊂ Ω n ( P ) . For any A ∈ Ω n ( P ) , by (2.10), it follows

A = G diag ( λ , 0 n − 1 ) G − 1 , P = G diag ( 1 , 0 n − 1 ) G − 1 , λ ∈ C .

Since A = λ P , then Ω n ( P ) ⊂ { λ P ∣ λ ∈ C } . □

By [34,35] or [15, Theorem 1.1], it has the following lemma.

Lemma 2.5

Let P ∈ C be idempotent. Then r ( P ) = t r ( P ) .

3 Main results

For a given idempotent matrix P ∈ C n × n , we assume

(3.1) M 1 = { A ∈ Ω n ( P ) ∣ A 2 − x A ≠ 0 , for any x ∈ C } , M 2 = { A ∈ Ω n ( P ) ∣ A 2 = 0 } , M 3 = { A ∈ Ω n ( P ) ∣ there exist some λ ∈ C ∗ such that A 2 = λ A , and r ( A ) < r ( P ) } , M 4 = { A ∈ Ω n ( P ) ∣ there exist some λ ∈ C ∗ such that A 2 = λ A , and r ( A ) = r ( P ) } .

Example 3.1

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , in view of Lemma 2.5 yields r ( P ) = t r ( P ) = r ≥ 2 . By applying Lemma 2.3, there exists an invertible matrix G such that P = G diag ( I r , 0 n − r ) G − 1 . Let A 1 = G diag ( 1 , − I r − 1 , 0 n − r ) G − 1 ; A 2 = G diag 0 0 1 0 , 0 n − 2 G − 1 ; A 3 = G diag ( λ , 0 n − 1 ) G − 1 and A 4 = G diag ( λ I r , 0 n − r ) G − 1 , λ ∈ C ∗ . Then A j P = P A j = A j , j = 1 , 2 , 3 , 4 .

As A 1 2 = P , and A 1 2 − x A 1 = G diag ( ( 1 − x ) , ( 1 + x ) I r − 1 , 0 n − r ) G − 1 ≠ 0 n , for any x ∈ C , by (3.1), it has A 1 ∈ M 1 .

Since A 2 is nilpotent, by (3.1), it has A 2 ∈ M 2 .

As A 3 2 = λ A 3 and r ( A 3 ) = 1 < r ( P ) , by (3.1), it has A 3 ∈ M 3 .

However, A 4 2 = λ A 4 and r ( A 4 ) = r ( P ) = r , by (3.1), it follows A 4 ∈ M 4 .

Example 3.1 indicates that, for a given idempotent matrix P with t r ( P ) = r ( P ) = r ≥ 2 , M j in (3.1) is not empty, j = 1 , 2 , 3 , 4 .

Theorem 3.1

For a given idempotent matrix P ∈ C n × n , it has

  1. if r ( P ) = r = 1 , then Ω n ( P ) = { λ P ∣ λ ∈ C } = M 4 ∪ { 0 } .

  2. if r ( P ) = r ≥ 2 , then M j ≠ ∅ , j = 1 , 2 , 3 , 4 and

    (3.2) Ω n ( P ) = M 1 ∪ M 2 ∪ M 3 ∪ M 4 , M j ∩ M l = ∅ , j ≠ l .

    (3.3) Ω n ( P ) = ( M 1 ∪ M 2 ∪ M 3 ) ∪ M 4 a n d ( M 1 ∪ M 2 ∪ M 3 ) ∩ M 4 = ∅ .

Proof

If r ( P ) = r = 1 , it follows (1) from (3.1) and Lemma 2.4 immediately.

The proof of (2). At the moment, as r ( P ) = r ≥ 2 , by Example 3.1, it has M j ≠ ∅ , j = 1 , 2 , 3 , 4 . Combining with (3.1) yields M j ∩ M l = ∅ , j ≠ l ; j , l = 1 , 2 , 3 , 4 . It is clear that M 1 ∪ M 2 ∪ M 3 ∪ M 4 ⊂ Ω n ( P ) .

For all A ∈ Ω n ( P ) , when A and A 2 are linear independent, that is, A 2 − x A ≠ 0 , for all by (3.1), it follows A ∈ M 1 ; when A 2 and A are linear dependent, there exists x 0 ∈ C such that A 2 = x 0 A . If x 0 = 0 , by (3.1), then A ∈ M 2 ; if x 0 ( = λ ) ≠ 0 , then A 2 = λ A . As A P = P A = A , it follows r ( A ) ≤ r ( P ) . When r ( A ) < r ( P ) , combining with (3.1) yields A ∈ M 3 . When r ( A ) = r ( P ) , it follows A ∈ M 4 .

Consequently, Ω n ( P ) ⊂ M 1 ∪ M 2 ∪ M 3 ∪ M 4 . The proof is completed.□

From Theorem 3.1, it is easy to see the following result.

Corollary 3.1

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A ∈ Ω n ( P ) with A ∈ M 1 . Then for any expression as A 2 = α A + β P , it has β ≠ 0 .

Theorem 3.2

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A ∈ Ω n ( P ) . Then A ∈ M 4 if and only if there exists λ ∈ C ∗ , such that A = λ P . And then,

(3.4) M 4 = { λ P ∣ λ ∈ C ∗ } .

Proof

First, if A = λ P for some λ ∈ C ∗ , then it is obvious that r ( A ) = r ( P ) and A 2 = A ( λ P ) = λ A . Hence, A ∈ M 4 from (3.1).

Second, if A ∈ M 4 , by (3.1), A 2 = λ A , for some λ ∈ C ∗ and r ( A ) = r ( P ) , by applying (2.10), we obtain

(3.5) A 2 = G diag ( A 0 2 , 0 n − r ) G − 1 = λ A = G diag ( λ A 0 , 0 n − r ) G − 1 , P = G diag ( I r , 0 n − r ) G − 1 .

Hence, A 0 2 = λ A 0 and r ( A 0 ) = r ( A ) = r ( P ) = r . Then it follows 1 λ A 0 2 = 1 λ A 0 with 1 λ A 0 is invertible. By applying Lemma 2.3, there exists an invertible matrix G 1 ∈ C r × r such that 1 λ A 0 = G 1 I r G 1 − 1 , combining with (3.5) yields A = λ G diag ( I r , 0 n − r ) G − 1 = λ P .

This implies, A ∈ M 4 if and only if there exists some λ ∈ C ∗ such that A = λ P . Thus, the proof is completed.□

Corollary 3.2

For a given idempotent matrix P ∈ C n × n , then M 4 ˜ = M 4 ∪ { 0 } is a subalgebra of C n × n with one dimension.

Proof

Let A , B ∈ M 4 ˜ , if one of A , B , say B is 0, then A + B , k A , A B ∈ M 4 ˜ , for all k ∈ C .

Now if A ≠ 0 and B ≠ 0 , by Theorem 3.2, there exist λ , μ ∈ C ∗ such that A = λ P , B = μ P , and hence, A B = λ μ P ∈ M 4 . Moreover, for any a , b ∈ C , we have a A + b B = ( a λ + b μ ) P ∈ M 4 ⊂ M 4 ˜ .

In all, M 4 ˜ is a subalgebra of C n × n with one dimension.□

As special classes of Ω n ( P ) , A 2 = ± A , A 2 = ± P have close relationships with M 4 .

Corollary 3.3

For a given idempotent matrix P ∈ C n × n , let A ∈ Ω n ( P ) . Then:

  1. When r ( P ) = 1 ,

    1. if A 2 = ± A , then A = ± P .

    2. If A 2 = P , then A = ± P .

    3. If A 2 = − P , then A = ± i P .

  2. When r ( P ) = r ≥ 2 ,

    1. if A 2 = ± A , then A ∈ M 3 ∪ M 4 .

    2. If A 2 = ± P , then A ∈ M 1 ∪ M 4 .

Proof

(1) When r ( P ) = 1 , by (1) in Theorem 3.1, we have A = λ P , for some λ ∈ C ∗ . If A 2 = ± A (resp. A 2 = P , resp. A 2 = − P ), then λ A = λ 2 P = A 2 = ± A (resp. λ 2 P = A 2 = P , resp. λ 2 P = A 2 = − P ), so λ = ± 1 (resp. λ = ± 1 , resp. λ = ± i ).

(2) When r ( P ) = r ≥ 2 .

If A 2 = ± A , for r ( A ) < r ( P ) (resp. r ( A ) = r ( P ) ), by (3.1), it has A ∈ M 3 (resp. A ∈ M 4 ).

If A 2 = P , by (3.1) and (2.10), it follows

G diag ( I r , 0 n − r ) G − 1 = P = A 2 = G diag ( A 0 2 , 0 n − r ) G − 1 ,

that is, A 0 2 = I r , By applying Lemma 2.2, there exists an invertible matrix G 1 such that A 0 = G 1 diag ( I s , − I r − s ) G 1 − 1 , 0 ≤ s ≤ r . Therefore,

A = G diag ( G 1 , I n − r ) diag ( diag ( I s , − I r − s ) , 0 n − r ) ( diag ( G 1 , I n − r ) ) − 1 G − 1 , P = G diag ( G 1 , I n − r ) diag ( I r , 0 ) ( diag ( G 1 , I n − r ) ) − 1 G − 1 .

Now let W = G diag ( G 1 , I n − r ) , then

(3.6) A = W diag ( I s , − I r − s , 0 n − r ) W − 1 , P = W diag ( I r , 0 n − r ) W − 1 , 0 ≤ s ≤ r .

For s = 0 (resp. s = r ), by (3.6), we obtain A = − P (resp. A = P ), so A ∈ M 4 . For 0 < s < r , by (3.1) and (3.6), we obtain A ∈ M 1 .

If A 2 = − P , by (2.10), it follows − P = A 2 = G diag ( A 0 2 , 0 n − r ) G − 1 , and hence A 0 2 = − I r , and by applying Lemma 2.3, there exists an invertible matrix G 1 such that A 0 = G 1 diag ( i I s , − i I r − s ) G 1 − 1 , 0 ≤ s ≤ r . Similar to (3.6), let W = G diag ( G 1 , I n − r ) , we obtain

(3.7) A = i W diag ( I s , − I r − s , 0 n − r ) W − 1 , P = W diag ( I r , 0 n − r ) W − 1 , 0 ≤ s ≤ r .

For s = 0 (resp. 0 < s < r , resp. s = r ), from (3.1) and (3.7), it has A ∈ M 4 (resp. A ∈ M 1 , resp. A ∈ M 4 ).□

If P ( = P 2 ) ∈ C n × n for A ∈ M 4 , from now on, we all assume A = λ P ( ≠ 0 ) , λ ∈ C ∗ .

Theorem 3.3

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A = λ P ∈ M 4 and κ ( A ) be determined by (1.5) and (1.6). Then

(3.8) A 2 = α A + ( λ 2 − λ α ) P , ∀ α ∈ C ,

(3.9) = λ − β λ A + β P , ∀ β ∈ C ,

(3.10) A k + 1 = x A + ( λ k + 1 − λ x ) P , ∀ x ∈ C ,

(3.11) = λ k − y λ A + y P , ∀ y ∈ C .

(3.12) κ ( A ) = z A + ( κ ( λ ) − λ z ) P , ∀ z ∈ C ,

(3.13) = 1 λ ( κ ( λ ) − ω ) A + ω P , ∀ ω ∈ C .

Proof

Since A = λ P ∈ M 4 , then A 2 = λ A . By Lemma 2.1, λ is unique. Combining with (1.5) and (1.6) yields κ ( A ) = κ ( λ ) P . For any α , β , x , y ∈ C , it follows

α A + ( λ 2 − λ α ) P = ( α λ + λ 2 − λ α ) P = λ 2 P = λ A = A 2 ,

λ − β λ A + β P = λ − β λ λ P + β P = λ 2 P = A 2 .

x A + ( λ k + 1 − x λ ) P = ( x λ + λ k + 1 − x λ ) P = λ k ( λ P ) = λ k A = A k + 1 ,

λ k − y λ A + y P = ( λ k + 1 − y + y ) P = λ k + 1 P = A k + 1 .

z A + ( κ ( λ ) − λ z ) P = ( λ z + κ ( λ ) − λ z ) P = κ ( λ ) P = κ ( A ) ,

1 λ ( κ ( λ ) − ω ) A + ω P = ( κ ( λ ) − ω + ω ) P = κ ( λ ) P = κ ( A ) .

Consequently, it follows (3.8)–(3.13).□

Theorem 3.4

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A = λ P ∈ M 4 and B is a linear combination of A and P. Then A B A = A if and only if

(3.14) B = ω A + 1 λ ( 1 − λ 2 ω ) P = 1 λ 2 ( 1 − λ μ ) A + μ P , for s o m e ω , μ ∈ C .

Proof

The proof for the sufficiency is trivial. If B = ω A + 1 λ ( 1 − λ 2 ω ) P , then A B A = ( λ P ) λ ω + 1 λ − λ ω P ( λ P ) . If B = 1 λ 2 ( 1 − λ μ ) A + μ P , then

A B A = ( λ P ) 1 λ 2 ( 1 − λ μ ) A + μ P ( λ P ) = λ P = A .

The proof for the necessity. Let B = ω A + μ P , then A B A = ( λ P ) ( ( λ ω + μ ) P ) ( λ P ) = λ 2 ( λ ω + μ ) P = A = λ P , hence λ 2 ω + λ μ = 1 . Then μ = 1 λ ( 1 − λ 2 ω ) , ω = 1 λ 2 ( 1 − λ μ ) , so (3.14) follows.□

If r ( P ) = r ≥ 2 , A ∈ M 4 , by (3.8) and (3.9), there are infinite kinds of expressions for A with the form as (1.1), so are the ones for A k + 1 , κ ( A ) and g -inverse of A as a linear combination of A and P .

If r ( P ) = 1 , A ∈ Ω n ( P ) , by (1) in Theorem 3.1, there always exists λ ∈ C such that A = λ P .

Then combining with the proofs of Theorems 3.3 and 3.4, we are led to

Corollary 3.4

For a given idempotent matrix P ∈ C n × n with r ( P ) = r = 1 , let A ( = λ P ) ∈ Ω n ( P ) . Then (3.8)–(3.14) hold.

It indicates that the properties of A = λ P ∈ M 4 for a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , also work for the case of r ( P ) = r = 1 .

Theorem 3.5

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , then the expression for A ∈ Ω n ( P ) with the form as (1.1)is unique if and only if A ∈ M 1 ∪ M 2 ∪ M 3 if and only if A ∉ M 4 .

Proof

The proof for the sufficiency. Let A ∈ M 1 ∪ M 2 ∪ M 3 . If there is another expression for A , A 2 = α ′ A + β ′ P different from (1.1), then

(3.15) ( α ′ − α ) A = ( β ′ − β ) P .

When A ∈ M 1 , if α ≠ α ′ , by (3.15), it has A = x 0 P , where x 0 = β ′ − β α ′ − α ∈ C ∗ , contradicting with (3.1), hence, α = α ′ and β = β ′ .

When A ∈ M 2 , then A 2 = 0 , by (1.1), α A = − β P . If α ≠ 0 , then A = β α P ≠ 0 , so 0 = A 2 = β α 2 P ( ≠ 0 ) , which gives a contradiction. Hence, α = β = 0 .

When A ∈ M 3 , by (3.1), it has A 2 = λ A = α A + β P ; if β = 0 , then λ A = A 2 = α A , from Lemma 2.1, λ = α . If β ≠ 0 , as ( λ − α ) A = β P , then r ( A ) = r ( P ) , namely, A ∈ M 4 , this contradicts with A ∈ M 3 . Hence, β = 0 and α = λ .

The proof for the necessity. Let the expression (1.1) for A be unique, if A ∉ M 1 ∪ M 2 ∪ M 3 , by applying (3.3), it has A ∈ M 4 . Combining with (3.8) and (3.9) yields the expression with the form as (1.1) is unique, this is a contradiction. Therefore, A ∈ M 1 ∪ M 2 ∪ M 3 .□

Theorem 3.6

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A ∈ Ω n ( P ) . Then (2.9) holds if and only if A ∈ M 1 ∪ M 2 ∪ M 3 if and only if A ∉ M 4 .

Proof

The proof for the necessity. If (2.9) holds and A ∉ M 1 ∪ M 2 ∪ M 3 , by (3.3), it follows A ∈ M 4 . By applying (3.8) and (3.9), there exists α ≠ α ′ ∈ C such that α A + ( λ 2 − λ α ) P = α ′ A + ( λ 2 − λ α ′ 2 ) P ( = A 2 ) , a contradiction.

The proof for the sufficiency. Let A ∈ M 1 ∪ M 2 ∪ M 3 and b 1 A + c 1 P = b 2 A + c 2 P , that is, ( b 1 − b 2 ) A = ( c 2 − c 1 ) P . If b 1 ≠ b 2 , then A = c 2 − c 1 b 1 − b 2 P ( ≠ 0 ) . If follows A ∈ M 4 from (3.4), a contradiction. Hence, b 1 = b 2 and c 1 = c 2 .

On the other side, if b 1 = b 2 , c 1 = c 2 , (2.9) holds naturally.□

If r ( P ) = r ≥ 2 , (2.9) is equivalent to A ∈ Ω n ( P ) and P are linear independent. By Theorems 3.1, 3.2, and 3.6, we are led to the following corollary.

Corollary 3.5

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A ∈ Ω n ( P ) . Then:

  1. The coefficients of expression for A k + 1 , κ ( A ) as a linear combination of A and P is unique (resp. not unique) if and only if A ∈ M 1 ∪ M 2 ∪ M 3 (resp. A ∈ M 4 ) if and only if A and P are linear independent (resp. linear dependent).

  2. When B = ω A + μ P and A B A = A , then ω and μ are unique (resp. nonuniqueness) if and only if A ∈ M 1 ∪ M 2 ∪ M 3 (resp. A ∈ M 4 ) if and only if A and P are linear independent (resp. linear dependent).

Example 3.2

Let P = 1 4 1 − 9 6 − 1 1 2 − 1 − 3 6 , A = 1 2 2 − 16 10 0 − 6 6 1 − 17 14 , B = 1 2 1 − 8 5 1 − 10 7 2 − 19 13 , from [13, Example 1], P 2 = P , r ( P ) = 2 , A P = P A = A , B P = P B = B , and ( A − 2 P ) ( A − 3 P ) = A 2 − 5 A + 6 P = 0 , ( B − P ) 2 = B 2 − 2 B + P = 0 , and hence, A , B ∈ Ω 3 ( P ) . As A , B and P are linear independent, by applying Corollary 3.5, we see that the expression A 2 = 5 A − 6 P , B 2 = 2 B − P are unique. It follows A , B ∈ M 1 from Corollary 3.1.

Example 3.3

Let A = 0 0 0 0 0 − 1 − 3 1 0 0 1 0 0 0 1 0 , P = 0 0 0 0 0 1 1 − 1 0 0 1 0 0 0 1 0 , by [13, Example 3], it has P 2 = P , A P = P A = A , A 2 = P , which means A is generalized involutory. Now combining with r ( P ) = 2 and Corollary 3.3 yields A ∈ M 1 ∪ M 4 . As A and P are linear independent, by Corollary 3.5, A ∉ M 4 , and hence, A ∈ M 1 .

Example 3.4

Let P = 1 − 1 0 0 0 0 0 0 − 3 4 2 − 1 − 6 7 2 − 1 , A = 0 0 0 0 0 0 0 0 5 − 6 − 2 1 5 − 6 − 2 1 , by [13, Example 4], it follows P 2 = P , r ( P ) = 2 > 1 = r ( A ) , A P = P A = A . As A ( A + P ) = 0 , then A 2 = − A , which means A is skew idempotent. From (3.1) and Corollary 3.5, we obtain A ∈ M 3 .

Example 3.5

Let

(3.16) P = 1 6 0 − 1 − 2 2 0 − 2 − 4 − 8 0 4 8 4 0 0 0 6 , A 1 = 1 6 6 1 2 − 2 0 8 4 8 0 − 4 − 2 − 4 0 0 0 0 , A 2 = 1 2 4 1 2 − 2 0 6 4 8 0 − 4 − 4 − 4 0 0 0 − 2 .

In view of Uç et al. [14, Example 2], P 2 = P , A 1 , A 2 are quadratic matrices with ( A 1 − I ) A 1 = A 1 2 − A 1 = 0 , ( A 2 − 2 I ) ( A 2 + I ) = A 2 2 − A 2 − 2 I = 0 , respectively. Now by (3.16) and [14, Example 2], it follows

(3.17) A = a 1 A 1 + a 2 A 2 = − P ( = − A 2 ) , A P = P A = A , ( a 1 , a 2 ) = ( − 2 , 1 ) ;

(3.18) A = a 1 A 1 + a 2 A 2 = P ( = A 2 ) , A P = P A = A , ( a 1 , a 2 ) = ( 2 , − 1 ) .

From (3.17), (3.18), and Theorem 3.2, numerical examples in the study by Uç et al. [14, Example 2] have A = a 1 A 1 + a 2 A 2 ∈ M 4 .

Example 3.6

Let matrices A 1 , A 2 , and P be same as (2.1) in Example 2.1, by (2.2)–(2.4) or [14, Example 3], it follows:

(3.19) A = a 1 A 1 + a 2 A 2 = 3 P = 1 3 A 2 , A P = P A = A , ( a 1 , a 2 ) = − 3 4 , − 3 2 ,

(3.20) A = a 1 A 1 + a 2 A 2 = − P ( = − A 2 ) , A P = P A = A , ( a 1 , a 2 ) = 1 4 , 1 2 ,

which indicates that the numerical examples in the study by Uç et al. [14, Example 3] have A = a 1 A 1 + a 2 A 2 ∈ M 4 .

Examples 3.5 and 3.6 tell us that the subset M 4 of Ω n ( P ) plays a very significant role in the recent researches on generalized quadratic matrices.

4 Some discussions on the related questions

In this section, we make further discussions on some basic properties studied by Farebrother and Trenkler [1] and some conclusions obtained by other authors [6,7,11–14,21] recently.

4.1 A equivalent expression for generalized quadratic matrices

Different from the previous study [1,3,4,10], the generalized quadratic matrix discussed in the previous studies [5,6,8,9,11–14,21] are defined in another way and has notation similar to the quadratic matrix in [2]:

For a given idempotent matrix P ∈ C n × n , if

(4.1) ( A − d P ) ( A − e P ) = 0 , A P = P A = A ,

then A is called as the { d , e } generalized quadratic matrix with respect to P , denoted by A ∈ Ω n ( P ; d , e ) .

Theorem 4.1

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A ∈ C n × n with A P = P A = A . Then

(4.2) A 2 = α A + β P ⇔ ( A − d P ) ( A − e P ) = 0 , α = d + e , β = − d e ; d , e = 1 2 ( α ± α 2 + 4 β ) .

Proof

By (2.10),

A 2 − α A − β P = G diag ( A 0 2 − α A 0 − β I r , 0 ) G − 1 ,

( A − d P ) ( A − e P ) = G diag ( ( A 0 − d I r ) ( A 0 − e I r ) , 0 ) G − 1 ,

then

(4.3) A 2 = α A + β P ⇔ x 2 − α x − β is a annihilating polynomial of A 0 .

(4.4) ( A − d P ) ( A − e P ) = 0 ⇔ ( x − d ) ( x − e ) is a annihilating polynomial of A 0 .

Due to the uniqueness of factorization of polynomial, x 2 − α x − β = ( x − d ) ( x − e ) , where α = d + e , β = − d e ; d , e = 1 2 ( α ± α 2 + 4 β ) . Then it follows (4.2) by (4.3) and (4.4).□

No matter which kind of definitions of generalized quadratic matrix, the condition A P = P A = A is necessary.

Example 4.1

Let P = 0 1 0 0 1 0 0 − 2 0 , A = 0 1 0 0 1 0 0 2 2 , then P 2 = P , A 2 = 2 A − P , by (4.2), α = 2 , β = − 1 , then d = e = 1 2 ( 2 ± 2 2 − 4 ) = 1 , namely, ( A − P ) ( A − P ) = ( A − P ) 2 . In fact, ( A − P ) 2 = 0 0 0 0 0 0 0 4 2 2 ≠ 0 . Therefore, the condition “ A P = P A = A ” for (4.2) is necessary.

Theorem 4.1 indicates expressions (1.1) and (4.1) are equivalent, and the coefficients are determined by (4.2) mutually. In view of Theorems 3.3, 3.5, and 4.1, we are led to the following corollary.

Corollary 4.1

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A ∈ Ω n ( P ) . Then:

( 1.1 ) i s u n i q u e ( r e s p . n o t u n i q u e ) ⇔ A ∈ M 1 ∪ M 2 ∪ M 3 ( r e s p . A ∈ M 4 ) ⇔ ( 4.1 ) i s u n i q u e ( r e s p . n o t u n i q u e ) .

Corollary 4.2

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 2 , let A = λ P ∈ M 4 , and (4.1) holds. Then d = λ or e = λ .

Proof

By (4.1), 0 = ( A − d P ) ( A − e P ) = ( λ − d ) ( λ − e ) P . Then d = λ or e = λ . □

From (1) in Theorem 3.1, if r ( P ) = r = 1 , there always has A = λ P ∈ Ω n ( P ) , by applying Corollary 3.4 yields the conclusion of Corollary 4.2 follows.

Now without loss of generality, if A = λ P ∈ M 4 , we always denote e = λ , then by (4.1), we are led to the following corollary.

Corollary 4.3

For a given idempotent matrix P ∈ C n × n with r ( P ) = r ≥ 1 , if A = λ P ∈ M 4 . Then A ∈ Ω n ( P ; d , e ) , for all d ∈ C , e = λ .

Uç et al. [14, Examples 2 and 3] showed us, A in Example 3.5 satisfying (3.17) (resp. (3.18)) belongs to Ω 4 ( P ; 1 , − 1 ) (resp. Ω n ( P ; 3 , − 1 ) ). However, by Corollary 4.3, a 1 A 1 + a 2 A 2 = A satisfying (3.17) (resp. (3.18)) should belong to Ω 4 ( P ; d , − 1 ) , Ω 4 ( P ; d , 1 ) (resp. Ω 4 ( P ; d , 3 ) , Ω 4 ( P ; d , − 1 ) ), for all d ∈ C .

For Ω n ( P ; d , e ) with d ≠ e , [11] used the notation C A = 1 e − d ( A − d P ) . If A = λ P ∈ M 4 , from Corollary 4.3, A ∈ Ω n ( P ; d , e ) , for all d ∈ C , λ = e . Consequently, C A = 1 e − d ( A − d P ) = 1 λ − d ( λ − d ) P = P , which means C A is independent of the choice of d ∈ C , and hence, the results of [11, Theorem 8] is still correct. Meanwhile, the conclusions in [6,8,9], which used C A = 1 e − d ( A − d P ) = P as a basic tool, is correct as well.

By (4.2), d − e = 4 β + α 2 . Then if A ∈ Ω n ( P ; d , e ) , d ≠ e ⇔ η 2 = 4 β + α 2 ≠ 0 .

When A = λ P ∈ M 4 , by applying Corollary 4.3, the coefficients of A as a generalized quadratic matrix are arbitrary. In 2013, Özdemir and Petik [7] discussed quadratic matrices by using the results of [6,12] when P = I n as basic tools.

4.2 Further discussions on generalized quadraticity of linear combination of two generalized quadratic matrices

For Q 1 , Q 2 ∈ Ω n ( P ) , Farebrother and Trenkler [1, Theorem 10] gave out some sufficient conditions such that Q 1 + Q 2 ∈ Ω n ( P ) .

Proposition 4.1

(See [1, Theorem 10]) Let Q 1 , Q 2 ∈ Ω n ( P ) such that

Q 1 2 = α Q 1 + β P , Q 2 2 = φ Q 2 + ψ P , Q 1 P = P Q 1 = Q 1 , Q 2 P = P Q 2 = Q 2 .

Then each set of the following conditions is sufficient for Q 1 + Q 2 ∈ Ω ( P ) :

  1. Q 1 Q 2 + Q 2 Q 1 = μ Q 1 + μ Q 2 + ν P , for some numbers μ and ν , α = φ ;

  2. Q 1 Q 2 + Q 2 Q 1 = μ P , for some numbers μ , α = φ ;

  3. Q 1 Q 2 = Q 1 , Q 2 Q 1 = Q 1 , φ = α + 2 ;

  4. Q 1 Q 2 = Q 1 , Q 2 Q 1 = Q 2 , α = φ ;

  5. Q 1 Q 2 = Q 2 , Q 2 Q 1 = Q 1 , α = φ ;

  6. Q 1 Q 2 = Q 2 , Q 2 Q 1 = Q 2 , α = φ + 2 .

In 2015, Petik et al. [12] considered some more general case.

Proposition 4.2

(See [12, Theorem 2.1]) Let Q j ∈ Ω n ( P ; d j , e j ) with d j ≠ e j , j = 1 , 2 and P = P 2 ∈ C n × n . Consider the linear combination of the form

(4.5) Q = γ 1 Q 1 + γ 2 Q 2 , γ 1 , γ 2 ∈ C ∗ .

Then:

  1. The matrix Q of the form (4.5) is a generalized ( λ , μ ) quadratic matrix with respect to P if and only if any of the following sets of additional conditions holds if Q 1 Q 2 = Q 2 Q 1 , Q 1 ≠ e 1 P , Q 2 ≠ e 2 P , and ( d 2 − e 2 ) ( Q 1 − e 1 P ) ≠ ( d 1 − e 1 ) ( Q 2 − e 2 P ) :

    1. c j ( c j + 2 c 3 − λ − μ ) + ( c 3 − λ ) ( c 3 − μ ) = 0 , j = 1 , 2 , P 1 P 2 = 0 , and ( c 3 − λ ) ( c 3 − μ ) ( P 1 + P 2 − P ) = 0 .

    2. 2 c j + c l + 2 c 3 − λ − μ = 0 , c j ( c j + 2 c 3 − λ − μ ) + ( c 3 − λ ) ( c 3 − μ ) = 0 , P 1 P 2 = P l , P 1 P 2 ≠ P j , and ( c 3 − λ ) ( c 3 − μ ) ( P j − P ) = 0 , ( j , l ) = ( 1 , 2 ) or ( j , l ) = ( 2 , 1 ) ,

    3. c j ( c j + 2 c 3 − λ − μ ) + ( c 3 − λ ) ( c 3 − μ ) = 0 , P 1 P 2 ≠ P j , j = 1 , 2 , P 1 P 2 ≠ 0 , and P 1 P 2 ≠ 0 , and ( c 3 − λ ) ( c 3 − μ ) ( P 1 + P 2 − P ) = 2 c 1 c 2 P 1 P 2 .

  2. The matrix Q of the form (4.5) is a { λ , μ } generalized-quadratic matrix with respect to P if and only if c 1 + c 2 + 2 c 3 = λ + μ and ( c 3 − λ ) ( c 3 − μ ) P − c 1 c 2 ( P 1 + P 2 − P 1 P 2 − P 2 P 1 ) = 0 if Q 1 Q 2 ≠ Q 2 Q 1 , where c j = γ j ( d j − e j ) , P j = ( d j − e j ) − 1 ( Q j − e j P ) , j = 1 , 2 and c 3 = γ 1 e 1 + γ 2 e 2 .

In view of the analysis of [12, Theorem 1.1] yields that, the idempotent matrix P j = 1 d j − e j ( Q j − e j P ) determined by Q j ∈ Ω n ( P ; d j , e j ) ( d j ≠ e j ) is just the one studied in [1, Theorem 6].

Similar to [15], [12] discussed the generalized quadraticity in two case: Q 1 Q 2 = Q 2 Q 1 and Q 1 Q 2 ≠ Q 2 Q 1 .

When Q 1 ( = e 1 P ) ∈ M 4 , from Theorem 3.2 and Corollary 4.3, Q 1 ∈ Ω n ( P ; d 1 , e 1 ) , ∀ d 1 ∈ C , and for Q 2 ∈ Ω n ( P ) , it always has Q 1 Q 2 = Q 2 Q 1 . This implies that there does not exist Q 2 ∈ Ω n ( P ) such that Q 1 Q 2 ≠ Q 2 Q 1 , if Q 1 ∈ M 4 , so it couldnot satisfy the condition for (1) in [12, Theorem 2.1] (i.e., Proposition 4.2). Consequently, the generalized quadratic matrices in Examples 2.1, 3.5, and 3.6, which belong to M 4 havenot been discussed in the study by Petik et al. [12]. From the view of research methods, the constraint conditions required in the study by Petik et al. [12] for Q j ∈ Ω n ( P ; d j , e j ) , d j ≠ e j , j = 1 , 2 is reasonable.

On the basis of the aforementioned discussions, we are led to the generalized quadraticity of a nonzero linear combination of Q 1 , Q 2 ∈ Ω n ( P ) .

Theorem 4.2

For a given idempotent matrix P ∈ C n × n , let Q 1 , Q 2 ∈ Ω n ( P ) . Then:

  1. If r ( P ) = 1 , for all γ 1 , γ 2 ∈ C ∗ , then γ 1 Q 1 + γ 2 Q 2 ∈ Ω n ( P ) .

  2. If r ( P ) = r ≥ 2 , let Q j ∈ Ω n ( P ; d j , e j ) with d j ≠ e j , j = 1 , 2 , then there exist λ , μ ∈ C , γ 1 , γ 2 ∈ C ∗ , it has γ 1 Q 1 + γ 2 Q 2 ∈ Ω n ( P ; λ , μ ) if and only if Q 1 , Q 2 ∈ M 1 ∪ M 2 ∪ M 3 and satisfy the conditions (1) or (2) in [12, Theorem 2.1].

  3. If r ( P ) = r ≥ 2 , Q 1 ∈ M 4 or Q 2 ∈ M 4 , then for all γ 1 , γ 2 ∈ C ∗ , there exist λ , μ ∈ C such that Q = γ 1 Q 1 + γ 2 Q 2 ∈ Ω n ( P ; λ , μ ) .

Proof

(1) If r ( P ) = 1 , for all γ 1 , γ 2 ∈ C , by (1) in Theorem 3.1, there exist e 1 , e 2 ∈ C such that Q 1 = e 1 P , Q 2 = e 2 P . Then γ 1 Q 1 + γ 2 Q 2 = ( γ 1 e 1 + γ 2 e 2 ) P ∈ Ω n ( P ) = M 4 ∪ { 0 } .

(2) If r ( P ) = r ≥ 2 , for Q j ∈ M 1 ∪ M 2 ∪ M 3 , d j ≠ e j with γ 1 Q 1 + γ 2 Q 2 ∈ Ω n ( P ; λ , μ ) , it follows (1) and (2) from [12, Theorem 2.1].

If Q j satisfy the conditions for (1) or (2) in [12, Theorem 2.1], according to the aforementioned analysis, Q j ≠ e j P , j = 1 , 2 , and therefore, Q j ∈ M 1 ∪ M 2 ∪ M 3 follow by Theorem 3.1 and (3.3), j = 1 , 2 .

(3) If r ( P ) = r ≥ 2 , might as well we take Q 1 = e 1 P ∈ M 4 . If Q 2 = e 2 P ∈ M 4 , then in view of Corollary 4.3 yields γ 1 Q 1 + γ 2 Q 2 = ( γ 1 e 1 + γ 2 e 2 ) P ∈ Ω n ( P ; λ , μ ) , μ = γ 1 e 1 + γ 2 e 2 , with λ ∈ C . So we only consider Q 2 ∈ Ω n ( P ; d 2 , e 2 ) ∈ M 1 ∪ M 2 ∪ M 3 . By (4.2) and (4.1), Q 2 2 = ( d 2 + e 2 ) Q 2 − d 2 e 2 P . On the other hand,

Q 2 = ( γ 1 Q 1 + γ 2 Q 2 ) 2 = ( γ 1 e 1 P + γ 2 Q 2 ) 2 = γ 1 2 e 1 2 P + 2 γ 1 γ 2 e 1 Q 2 + γ 2 2 ( d 2 + e 2 ) Q 2 − γ 2 2 d 2 e 2 P = [ γ 2 2 ( d 2 + e 2 ) + 2 γ 1 γ 2 e 1 ] Q 2 + ( γ 1 2 e 1 2 − γ 2 2 d 2 e 2 ) P = [ γ 2 ( d 2 + e 2 ) + 2 γ 1 e 1 ] ( γ 2 Q 2 + γ 1 e 1 P − γ 1 e 1 P ) + ( γ 1 2 e 1 2 − γ 2 2 d 2 e 2 ) P = [ γ 2 ( d 2 + e 2 ) + 2 γ 1 e 1 ] Q − ( γ 1 2 e 1 2 + γ 1 γ 2 e 1 d 2 + γ 1 γ 2 e 1 e 2 + γ 2 2 d 2 e 2 ) P ,

namely,

(4.6) Q 2 = α Q + β P , α = γ 2 ( d 2 + e 2 ) + 2 γ 1 e 1 , β = − ( γ 1 2 e 1 2 + γ 1 γ 2 e 1 d 2 + γ 1 γ 2 e 1 e 2 + γ 2 2 d 2 e 2 ) .

Let λ , μ be the roots of x 2 − α x − β . Then λ , μ = 1 2 ( α ± α 2 + 4 β ) , and λ + μ = α , λ μ = 1 4 ( α 2 − α 2 − 4 β ) = β . By (4.6), ( Q − λ P ) ( Q − μ P ) = Q 2 − ( λ + μ ) Q + λ μ P = 0 , namely, Q = γ 1 Q 1 + γ 2 Q 2 ∈ Ω n ( P ; λ , μ ) . □

Let the idempotent matrix P and A ∈ C 3 × 3 be given by Example 3.2, in view of [13] yields ( A − 2 P ) ( A − 3 P ) = 0 , namely, A 2 = 5 A − 6 P and A ∈ M 1 . Let Q 1 = e 1 P ∈ M 4 , Q 2 = A , then there are unique numbers φ , ψ ∈ C such that Q 2 2 = φ Q 2 + ψ P , φ = 5 , ψ = − 6 . By (3.8), Q 1 2 = α Q 1 + ( e 1 2 − e 1 α ) P , for any α ∈ C , if α = 6 , then α ≠ 5 = φ , α ≠ 7 = φ + 2 , φ = 5 ≠ 8 = α + 2 . This indicates that Q 1 and Q 2 do not satisfy any sufficient conditions for Proposition 4.1 (i.e., [1, Theorem 10]).

Farebrother and Trenkler [1, Theorem 9] also showed some sufficient conditions for Q 1 Q 2 ∈ Ω n ( P ) , where Q 1 , Q 2 ∈ Ω n ( P ) . Now based on Theorem 4.2, we are led to the following result.

Theorem 4.3

For a given idempotent matrix P ∈ C n × n , let Q 1 , Q 2 ∈ Ω n ( P ) . Then:

  1. If r ( P ) = 1 , Q 1 + Q 2 , Q 1 Q 2 ∈ Ω n ( P ) .

  2. If r ( P ) = r ≥ 2 and Q j ∈ M 1 ∪ M 2 ∪ M 3 , where Q j ∈ Ω n ( P ; d j , e j ) with d j ≠ e j , j = 1 , 2 , then under the conditions of Proposition 4.1, it has Q 1 + Q 2 ∈ Ω n ( P ) .

  3. If r ( P ) = r ≥ 2 , and one of Q 1 , Q 2 belongs to M 4 , then Q 1 + Q 2 , Q 1 Q 2 ∈ Ω n ( P ) .

Proof

(1) follows by Theorem 3.1 and Corollary 3.2.

From Theorem 4.2, (2) holds if taking γ 1 = γ 2 = 1 .

If r ( P ) = r ≥ 2 and one of Q 1 , Q 2 belongs to M 4 , by (3) in Theorem 4.2, it follows Q 1 + Q 2 ∈ Ω n ( P ) , which is (2).

If Q 1 = e 1 P ∈ M 4 , let Q 2 2 = α 2 Q 2 + β 2 P . Then

( Q 1 Q 2 ) 2 = e 1 2 Q 2 2 = e 1 α 2 ( e 1 Q 2 ) + e 1 2 β 2 P = e 1 α 2 ( Q 1 Q 2 ) + e 1 2 β 2 P .

Since ( Q 1 Q 2 ) P = Q 1 Q 2 = P ( Q 1 Q 2 ) , by (1.1), Q 1 Q 2 ∈ Ω n ( P ) , which is (3).□

Acknowledgements

Thanks for the referees’ comments. Those comments are all valuable and very helpful for revising and improving our article, as well as the important guiding significance to our researches. The work is supported by National Natural Science Foundation of China (62372256), Fujian Provincial Natural Science Foundation (2021J011103, 2021J01985 and 2023J01997), and the Science and Technology Project of Putian City (2022SZ3001ptxy05).

  1. Conflict of interest: The authors state no conflicts of interest.

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Received: 2023-09-20
Revised: 2024-02-15
Accepted: 2024-02-16
Published Online: 2024-03-20

© 2024 the author(s), 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-2023-0186
Keywords for this article
generalized quadratic matrix; uniqueness; linear combination; rank
Creative Commons
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Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
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  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
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  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
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  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
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  61. Singularities of spherical surface in R4
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  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
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  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
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  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
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  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
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  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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