Maps Preserving Idempotence on Matrix Spaces

Abstract

Suppose is an arbitrary field. Let be the number of the elements of . Let be the space of all matrices over , let be the subset of consisting of all symmetric matrices, and let be the subset of consisting of all upper-triangular matrices. Let ; a map is said to preserve idempotence if is idempotent if and only if is idempotent for any and . In this paper, the maps preserving idempotence on , , and were characterized in case .

1. Introduction

Suppose is an arbitrary field and is the field of all complex numbers. Let be the number of the elements of . Let be the space of all matrices over , let be the subset of consisting of all symmetric matrices, and let be the subset of consisting of all upper-triangular matrices. For a linear subspace of , we denote by the subset of consisting of all idempotence. A map is said to preserve idempotence if is idempotent if and only if is idempotent for any and . Denote by the set of all maps from to such that Denote by the set of all maps from to such that For a matrix , we denote by the transpose of and by the inverse of . For any positive integer , we denote , , where denotes the usual direct sum of matrices. In particular, , and . Denote by the number of elements of . Denote , . Denote by the matrix with in the ()th entry and elsewhere. Denote . (Note: the dimension of can be known from the content.)

The problem studied in this paper belongs to preserver problems on spaces of matrices. In the recent years, some researchers have studied preserver problems on spaces of matrices [1–4]. Šemrl [5] characterized the bijective and continuous maps preserving idempotence on when . Liu and Zhao [6] improved the result of Šemrl by relaxing the bijectivity assumption to the surjectivity and also omitting the continuous assumption and the restriction on . Zhang [7] investigated the maps on preserving idempotence without the surjectivity assumption in the higher dimensions when is any field of characteristic not . But the homogeneous of Lemma 4 in his paper was not proved, and its original proof in [2] needs a nonzero scalar which is not equal to and . In [8] and [9] the authors characterized the maps preserving idempotence on and in case is any field of characteristic not and , respectively. Thus, the case of is not proved on , and . Clearly, it is more complex to characterize maps preserving idempotence without being homogeneous. In addition, Lemma of [7], Lemma of [8], and Lemma of [9] played an important role, but their original proof is no longer applicable in the case of . This paper will use a new method to prove these conclusions. The purpose of this paper is to prove the following theorems.

Theorem 1. Suppose is a field with and ; then is a map preserving idempotence on spaces of if and only if there exists an invertible matrix such that for every , where for some nonzero scalar of .

Theorem 2. Suppose is a field with and . Then is a map preserving idempotence on spaces of if and only if there exists an invertible matrix such that either for every or for every , where .

Theorem 3. Suppose is a field with ; then is a map preserving idempotence on spaces of if and only if there exists an invertible matrix such that for every or for every .

2. Some Lemmas

In order to prove the theorems, we require the following lemmas.

Lemma 4. Suppose is a field and . If is a map preserving idempotence on , or , or , then(i)is idempotent if and only if is idempotent for any , or ;(ii) is injective.

Proof. The proof of it is omitted, since it is very similar to that in [6].

Lemma 5. Suppose is a field and . If is a map preserving idempotence on , or , or , then there exists an invertible matrix such that for every .

Proof. The proof of it is omitted, since it is very similar to Lemmas and of [7], respectively.

Lemma 6. Suppose is a field and . If is a map preserving idempotence on , or , or , and , then for every of .

Proof. Since , , then by Lemma 5 we have , , . Thus .
For any , since , , then , . In view of the injectivity of and by direct calculation we can conclude that .

Lemma 7. Suppose is a field and , , and . If for every and , , , then for every , .

Proof. Let , , and ; then , . Since , , we can conclude that , . Let ; then . By direct calculation we can conclude that , where ; that is,Let , . By (3) we havewhere , . Since , , by (4) we have . By direct calculation we can conclude thatLet , , , , , , and ; then by (3) and (5) we haveSince , , then , . By (6) we have , ; this means that ; that is, . Similarly we have . Thus is independent of the choice of . Since , then . Similarly we can conclude that .

Lemma 8. Suppose is a field and , , and is a map preserving idempotence on or . If for every and , , , then for every .

Proof. Using by or , by , by , and by in the proof of Lemma 7, we can prove Lemma 8.

3. The Proof of Theorems 1 and 2

By Lemmas 4–8, we can prove Theorems 1 and 2 in similar way as in [8] and [9], respectively.

4. The Proof of Theorem 3

The proof of Theorem 3 is equivalent to the proof of the following four propositions.

Proposition 9. Suppose is a field and . If is a map preserving idempotence on spaces of , then there exists an invertible matrix such that for every .

Proof. In view of Lemma 6, it is obvious.

Proposition 10. Suppose is a field and . If is a map preserving idempotence on spaces of and for every , then there exists an invertible matrix such that for every or for every .

Proof. In view of Theorem 1, we can prove that there exists an invertible matrix such that for every . For any which satisfy , let , , , ; then , . Since , , , then , , . Let ; then . By direct calculation we can conclude that , where and are maps from to , and ; that is,Case  1. If , in view of the injectivity of we can conclude that ; that is, SincethenThus ; that is, For any , since thenSo , . In a similar way as above, we can conclude that Similarly, we have Let ; then we have For another nonzero scalar of , because , Similarly, we have Thus, for any , Case  2. If , then in a similar way as Case we can prove that

Proposition 11. Suppose and is a field and . If is a map preserving idempotence on spaces of and for every , then there exists an invertible matrix such that for every .

Proof.
Step  1. There exists an invertible matrix such that Let , , , ; then , Since , , , then , , . Let ; thus . This implies that , where is a map from to ; that is, For every , let , . Since , , then , . By direct calculation we can conclude thatSimilarly, for any , which satisfy that , by , , , , , we have This, together with (23), implies that is independent of the choice of and . Without loss of generality we also denote by . Let ; then similarity transformation by matrix preserves the above results, and In view of the injectivity of we can conclude that Step  2. Ifwhere , then For any , let , , , . By Lemma 7 we have , In a similar way as Step we can prove that where is a map from to , . By we can prove that is independent of the choice of . In view of we can conclude that .
Step  3. If , , , , then For any , let , , , . By Step we have , . In a similar way as Step we can prove that where and are maps from to , and . By we can prove that and are independent of the choice of . In view of , , we can conclude that , .
Step  4. If where and , then For any , let , , , . Then , . In a similar way as Step we can prove that where and are maps from to , , and . By we can prove that and are independent of the choice of . In view of , , we can conclude that , .

Proposition 12. Suppose , and is a field and . If is a map preserving idempotence on spaces of and for every , then there exists an invertible matrix such that for every .

Proof. In a similar way as the proof of Proposition 11 we can complete its proof.

Remark 13. The following example states that Theorem 1 is wrong in case .

Example 14. Suppose is a field and . Denote a map by for any , . It is easy to prove that or for every , where . Clearly, is a map preserving idempotence, but is not the form described in Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors show great thanks to the editor and referee for their careful reading of the paper and valuable comments which greatly improved the readability of the paper. This work was supported by the fund of Heilongjiang Education Committee (no. 12541605).