The uniqueness of expression for generalized quadratic matrices

: It is shown that the expression as = + A αA βP 2 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 su ﬃ cient and necessary conditions for the uniqueness of generalized quadratic matrices ’ expression are obtained. Finally, some related discussions about generalized quadratic matrices are also given


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 1

2
. And α | | stands for the modulus of ∈ α C. The symbols r A ( ) and tr 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 P Ω n ( ) as follows, for a given idempotent matrix P, By [1], ∈ A P Ω n ( ) is called as a generalized quadratic matrix with respect to P. Clearly 0 and ∈ P P Ω n ( ) is trivial.From now on, unless otherwise specified, both ∈ A P Ω n ( ) and P are nonzero.When = P I n , a subclass of P Ω n ( ) was investigated by Aleksiejczyk and Smoktynowicz [2], where A satisfies the quadratic equation , .Moreover, by (1.1), P Ω n ( ) contains the following subclasses (see [1,Introduction], ≠ P I n ): idempotent matrices 1, 0 generalized involutory matrices 0, 1 nilpotent matrices 0 0, 0 skew idempotent matrices 1, 0 generalized skew involutory matrices 0, 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 .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.
then for all ∈ + k Z , it has where (1.4) For any set of complex numbers κ κ κ , ,…, , Farebrother and Trenkler [1] also show that there is the finite order polynomial , where Ω .
Huang et al. [21] discussed the rank of matrices as of the form in (1.6).
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) Once α n is determined, we obtain the solution for β n by the identity = β βα n n .The characteristic equation belonging to (1.9) is given by Goldberg [29,Sec. 5.12] , where γ is one square root of , where γ is one square 3) can be expressed as follows: In view of studies by Goldberg [29] and Israel and Greville [30] In view of the study by Uç et al. [14,Example 3], both A 1 and A 2 are quadratic matrices with , respectively, and = P P .
2 By calculation, we obtain (2.2) From the study by Uç et al. [14,Example 3] ) and according to (1.1), , so (1) From (2.4), it follows Then combining with (1.4) and (1.10) yields ) it shows the coefficient for A 3 as a linear combina- tion of A and P couldnot be obtained from (1.4) . However, it follows from (1.12) in Proposition 1.
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).
The uniqueness of expression for generalized quadratic matrices  3 ∈ A P Ω n ( ) 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: We have to show This follows from ) by assumption, and thus, Seeking into the proof of Farebrother and Trenkler [1, Theorem 1] requires 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 P Ω n ( ) 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 be a scalar-idempotent matrix determined by λ.Then the scalar λ is unique.
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.

Main results
For a given idempotent matrix ∈ × P C n n , we assume Ω there exist some such that , and , Ω there exist some such that , and .
Example 3.1.For a given idempotent matrix By applying Lemma 2.3, there exists an invertible matrix G such that = P , and Theorem 3.1.For a given idempotent matrix , it follows (1) from (3.1) and Lemma 2.4 immediately.The proof of (2).At the moment, as ( ) , for all by (3.1), it follows ∈ A M ; 1 when A 2 and A are linear dependent, there exists ( ) .The proof is completed.□ The uniqueness of expression for generalized quadratic matrices  5 From Theorem 3.1, it is easy to see the following result.
Hence, = A λA 3, there exists an invertible matrix 1 , combining with (3.5) yields A M 4 if and only if there exists some ∈ * λ C such that = A λP. Thus, the proof is completed.□ Corollary 3.2.For a given idempotent matrix Proof.
), so , By applying Lemma 2.2, there exists an invertible matrix G 1 such that , and by applying Lemma 2.3, there exists an invertible matrix G 1 such that ), we obtain ( ) Consequently, it follows (3.
The uniqueness of expression for generalized quadratic matrices  7 Proof.The proof for the sufficiency is trivial.
The proof for the necessity.Let = + B ωA μP, then , ∈ 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.    , (2.9) is equivalent to ∈ A P Ω n ( ) and P are linear independent.By Theorems 3.1, 3.2, and 3.6, we are led to the following corollary.
The uniqueness of expression for generalized quadratic matrices  9 Examples 3.5 and 3.6 tell us that the subset M 4 of P Ω n ( ) plays a very significant role in the recent researches on generalized quadratic matrices.

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][12][13][14]21] recently.

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][12][13][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 Proof.By (2.10), ) .Therefore, the condition " = = AP PA A" for (4.2) is necessary.When = ∈ A λP 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.( ) such that

Further discussions on generalized quadraticity of linear combination of two generalized quadratic matrices
Then each set of the following conditions is sufficient for ( ): , for some numbers μ and = ν α φ , ; , for some numbers = μ α φ , ; In 2015, Petik et al. [12] considered some more general case.
. Then: The uniqueness of expression for generalized quadratic matrices  11 (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 e P

Theorem 3 . 4 .
For a given idempotent matrix ∈ × P C n n with = ≥ r P r 2, ( ) let = ∈ A λP M 4 and B is a linear combination of A and P. Then = ABA A if and only if by(3.15), it has = A x P,

2 , 1 4. 1 Example 3 . 4 .
As A B , and P are linear independent, by applying Corollary 3.5, we see that the expression = which means A is generalized involutory.Now combining with = r As A and P are linear independent, by Corollary 3.5, ∉ A M 4 , and hence, ∈ A M Let =

2. 3 Example 3 1 2
which means A is skew idempotent.From (3.1) and Corollary 3.5, we obtain ∈ A M are quadratic matrices with − Then it follows (4.2) by (4.3) and(4.4).□No matter which kind of definitions of generalized quadratic matrix, the condition = = AP PA A 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
Farebrother and Trenkler [1, Theorem 10]  gave out some sufficient conditions such that +

Proposition 4 . 2 . 2 .
(See [12, Theorem 2.1]) Let ∈ Q P d e Ω ; , Consider the linear combination of the form by (1) in Theorem 3.1, there always exists ∈ λ C such that = A λP. Then combining with the proofs of Theorems 3.3 and 3.4, we are led to 2different from (1.1), then ) if and only if A and P are linear independent (resp.linear dependent).
4.1.For a given idempotent matrix ∈ × is independent of the choice of ∈ d C, and hence, the results of [11, Theorem 8] is still correct.Meanwhile,