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).