Abstract and Applied Analysis

Volume 2014 (2014), Article ID 596756, 5 pages

http://dx.doi.org/10.1155/2014/596756

Research Article

Induced Maps on Matrices over Fields

Li Yang,¹ Xuezhi Ben,² Ming Zhang,² and Chongguang Cao²

¹Department of Foundation, Harbin Finance University, Harbin 150030, China

²School of Mathematical Science, Heilongjiang University, Harbin 150080, China

Received 21 October 2013; Revised 23 December 2013; Accepted 2 January 2014; Published 26 February 2014

Academic Editor: Chi-Keung Ng

Copyright © 2014 Li Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Suppose that is a field and are integers. Denote by the set of all matrices over and by the set . Let () be functions on , where stands for the set . We say that a map is induced by if is defined by . We say that a map on preserves similarity if , where represents that and are similar. A map on preserving inverses of matrices means for every invertible . In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.

1. Introduction

Suppose that is a field and are integers. Denote by the set of all matrices over and by the set . Let (, ) be functions on , where stands for the set . We say that map is induced by if is defined by

It is easy to see that induced map may not be linear or additive.

Example 1. Let be real field, , , , and , then induced by is

Example 2. The transposition map is not an induced map on .

Example 3. Let be a matrix; then is an induced map on if and only if is diagonal.

If is independent of the choices of and (i.e., , for every and ), then is said to be induced by the function , and denote by . Denote by the rank of matrix . We say that an induced map preserves rank-1 if whenever .

Preserver problem is a hot area in matrix and operator algebra; there are many results about this area. Kalinowski [1] showed that an induced map , where is a monotonic and continuous function of real field such that , preserves ranks of matrices if and only if it is linear. Furthermore, in [2], Kalinowski generalized the results in [1] by removing any restrictions on the map . In [3], Liu and Zhang characterized the general form of all maps induced by and preserving rank-1 matrices over a field. In particular, nonlinear maps preserving similarity were studied by Du et al. [4]. One can see [5–15] and their references for some background on preserver problems.

We say that a map on preserves similarity if

where represents that and are similar. A map on preserving inverses of matrices means for every invertible . In this paper, we describe the forms of induced map preserving similarity and inverses of matrices, respectively.

We end this section by introducing some notations which will be used in the following sections. Let be the diagonal matrix of order . is the matrix with 1 in the th entry and 0 elsewhere and is the identity matrix of order . Denote by the usual direct sum of matrices.

2. Induced Map Preserving Similarity of Matrices

In this section, we use the form of induced rank-1 preserver to describe forms of induced similarity preservers. Firstly, we need the following theorem from [3].

Lemma 4 (see [3, Corollary  1]). Suppose that is any field and are integers. Suppose that on is induced by such that . Then preserves rank-1 if and only if there exist invertible and diagonal and a multiplicative map on satifying such that

Lemma 5. Suppose that is any field, and is an integer with . If satisfies and , then there exists an invertible matrix such that

Proof. It is easy to see that there exists an invertible matrix such that

where and satisfy . From , we have

so that . Thus, , which implies that

for some invertible . Let ; then (6) turns into

This completes the proof.

Theorem 6. Let be a field and let be positive integers. Suppose that is a map on induced by such that . Then preserves similarity if and only if there exist an invertible and diagonal , , and an injective endomorphism of such that

Proof. The sufficiency is obvious. We will prove the necessary part by the following four steps.

Step  1. If there exists some and such that , then .

Proof of Step  1. For any , , since and , by Lemma 5, we have

Since preserves similarity, we derive

and thus,

Because of rank and , by Lemma 5, we have

Using preserves similarity and (13), one can obtain that

hence,

It follows from (13) and (16) that , that is, .

Step  2. If there exist some and such that , then .

Proof of Step  2. For , it follows from that . Thus,

Because of

one can obtain by using (17) that , and hence,

Thus, or . We complete the proof of this step by using the result of Step  1.

Step  3. If , then preserves rank-1.

Proof of Step  3. For any rank-1 matrix we have

and hence,

Thus, ; it follows from that .

Step  4. If , then there exist an invertible and diagonal , , and an injective endomorphism of such that

Proof of Step  4. Since , by Step  3 and Lemma 4, there exist invertible and diagonal and a multiplicative map on satisfying such that

Let

It is easy to see that . Since preserves similarity, we have that and are similar; further, . It follows from (23) that , thus,

This implies , hence, is an injective endomorphism of .

Set . Since , one obtains by using (23) that

Thus, . Letting , then and .

This completes the proof of Theorem.

3. Induced Map Preserving Inverses of Matrices

Theorem 7. Let be a field and let be positive integers. Suppose that is a map on induced by such that . Then preserves inverses of matrices if and only if there exist an invertible and diagonal , , and an injective endomorphism of such that

Proof. The sufficiency is obvious. We will prove the necessary part. For any , , and , since

by preserving inverses of matrices, we have

so that

Let ; then (30) turns into

Replacing by , then the above turns into

It follows from (30) and (33) that

Replacing by , then the above turns into

By , we have

In particular,

From , we have

In particular,

Multiplying by (31), we obtain by using (38) that

It follows from (35) and (38) that

This, together with (40), implies that

Hence, it follows from that

Let and ; we have

From

we have

so that

This, together with (39), implies

Similarly, ; hence,

It follows from

that

Hence,

This, together with (39) and (49), implies

Since

we have

so that

Hence,

For distinct and , since

we have

so that

This, together with (53), implies that

It follows from (35) and (61) that

Using this, together with (57), we obtain

where . Let ; and then

Let ; since preserves inverses of matrices, one can see that also preserves inverses of matrices. By , we obtain by using similar method to (35), (37), (44), and (53) that for any

This completes the proof.

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 referee for his/her careful reading of the paper and valuable comments which greatly improved the readability of the paper. Li Yang is supported by Vocational education institute in Heilongjiang Province “12th five-year development plan” The Guiding Function and Practice Research of Mathematical modeling in Advanced Mathematics Teaching of New Rise Financial Institutions’ (Grant no. GG0666). Chongguang Cao is supported by National Natural Science Foundation Grants of China (Grant no. 11371109).

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