Miscellaneous equalities for idempotent matrices with applications
-
Yongge Tian
Abstract
This article brings together miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices in one place with cogent introduction and references for further study. The author will present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix identities composed by the conventional operations of idempotent matrices, and uses of the block matrix method in the derivation of closed-form formulas for calculating the ranks of matrix expressions that are composed by idempotent matrices. The author also determines the maximum and minimum ranks of some matrix pencils composed by the products of matrices and their generalized inverses and uses the ranks to characterize algebraic performance of the matrix pencils.
1 Introduction
Throughout this article, let
A matrix X is called a
Recall that a square matrix A is said to be idempotent if
The purpose of this study is to provide a good coverage of idempotent matrix mathematics, from the development of the basic language to the collection of miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices and their conventional algebraic operations in one place with cogent introduction and references for further study. The remainder of this article is organized as follows. In Section 2, the author revisits some simple and well-known formulas for calculating the ranks of matrices in the literature and introduces the block matrix method (BMM) and its applications in establishing expansion formulas for calculating the ranks of block matrices, products of matrices, generalized inverses, etc. In Section 3, the author presents miscellaneous known and novel algebraic matrix identities composed by two and three idempotent matrices and discusses the applications of these identities in determining inverses and generalized inverses of matrix products. In Section 4, the author establishes a series of closed-form formulas for calculating the ranks of matrix expressions composed by idempotent matrices and uses the rank formulas to characterize a variety of fundamental properties of the matrix expressions. In Section 5, the author derives the upper and lower bounds for the ranks of some matrix pencils that involve the products of matrices and their generalized inverses. Some remarks and open problems on idempotent matrices and related issues are addressed in Section 6.
2 How to establish matrix rank equalities by the BMM
The rank of a matrix is a quite basic concept in linear algebra, which can be defined by different manners and can be calculated directly by transforming the matrix to certain row and/or column echelon forms. One of the most important applications of the rank of matrix is to describe singularity and nonsingularity of the matrix, as well as the dimension of the row or column space of the matrix. Thus, people would always be of interest in establishing various nontrivial analytical formulas for calculating the ranks of matrices under various assumptions. In fact, people have established a quite large number of equalities and inequalities for the ranks of matrices and widely used them in matrix analysis and applications. The author starts with a simple and best-known fact in linear algebra: the rank equalities
These results show that the rank of matrix has such an attractive feature that can be used to characterize matrix equalities and matrix set inclusions under general situations. To speak precisely, if certain closed-form formulas for calculating the rank of the difference
It has a long history in linear algebra to establish equalities and inequalities for the ranks of matrices from theoretical and applied points of view. Especially, there is a major route to derive several simple expansion formulas for calculating the ranks of matrix expressions by means of constructing various specific block matrices. Here, the author presents several well-known simple and interesting examples:
where A, B, X, and Y are the matrices of appropriate sizes. These rank equalities seem quite neat in form and are easy to understand. On the other hand, they can be used as matrix analytic tools to establish various complicated equalities and inequalities for the ranks of matrices and to characterize various fundamental properties of matrices, such as nullity, singularity, nonsingularity, idempotency, tripotency, and involution. A strong method of establishing these rank equalities is to construct a series of block matrices with the given matrices as follows:
and then to do some routine calculations of the ranks of the block matrices. Apparently, some of (2.1)–(2.8) occur in various textbooks in linear algebra and matrix theory [2,19,28,39]. The author hopes the above discussion gives the reader a familiar view of analytical expansion formulas for calculating the ranks of matrices, as well as the constructive use of the BMM that permits us to discover and prove these formulas.
In 1974, Marsaglia and Styan systematically approached in their study [19] a series of fundamental problems on the ranks of matrices and their generalized inverses and established a wide range of equalities and inequalities for the ranks of partitioned matrices and sums and products of matrices, and presented many practical applications of the rank equalities and inequalities in matrix theory and other fields. Here, the author presents several fundamental equalities and facts about the ranks of partitioned matrices and generalized inverses of the matrices in [19], which will be used in the sequel.
Lemma 2.1
[19] Let
the following rank equalities
(2.11)(2.12)(2.13)(2.14)hold for all
(2.15)(2.16)hold for all
If
(2.17)
Lemma 2.2
[19] Let
hold for all
It can be deduced from (2.18) and (2.19) that
and
which encompass the two famous Sylvester’s law
Lemma 2.3
[19] Let
holds for all
the following equivalent facts hold
Two new equalities for the rank of the partitioned matrix
Lemma 2.4
[35] Let
The range equality below holds
(2.26)The rank equalities below hold
(2.27)(2.28)
It can be imagined that one can derive many more nontrivial expansion formulas for calculating the ranks of matrix expressions through use of the BMM. In other words, it seems more natural to consider various specified block matrices and to extend (2.1)–(2.10) to general forms under various assumptions. The principal issue dealt with here is to what extent these rank formulas generalize to cases with multiple matrices. In the remaining part of this section, the author shows how to construct three block matrices from the general solutions of some fundamental linear matrix equations and derive three rank equalities associated with the solutions.
Theorem 2.5
Let
Then,
Proof
First construct a block matrix from X and Y as follows:
Then, it is easy to verify by EBMOs that
and from (2.29) that
Note that the block matrices
Combining these two expansion formulas leads to (2.30).□
Theorem 2.6
Let
Then,
Proof
Let N be as given in (2.31). Then, a routine deduction by EBMOs shows that
hold under (2.34). Combining these two expansion formulas leads to (2.35).□
Theorem 2.7
Let
Then,
Proof
Construct a block matrix from X, Y, AY, and BX as follows:
hold under (2.36). Combining these two rank equalities leads to (2.37).□
The preceding proofs of (2.30), (2.35), and (2.37) are elementary and straightforward by way of the BMM and EBMOs, which incontrovertibly seem to have some magic features that link the ranks of different matrices in a remarkable manner. Note that the matrix equations in (2.29), (2.34), and (2.36) are quite fundamental in matrix analysis and have been widely studied in theory and applications [20]. Under the assumptions of these equations, (2.30), (2.35), and (2.37) link the solutions of these matrix equations and their algebraic operations. In this situation, there is a very strong intrinsic mathematical motivation to establish concrete rank formulas from (2.30), (2.35), and (2.37) for various solutions of the matrix equations. As demonstrated in Section 4, the above BMM which we advocate can be used to deduce many interesting and delightful formulas and facts concerning the ranks of idempotent matrices and related issues.
Recall that the dimension of a finite-dimensional vector space (linear subspace) over the complex number field
holds for a family of linear subspaces
Concerning the rank of the product of two matrices, we have the following results.
Lemma 2.8
Let
Then,
Proof
It can be deduced from (2.40) and EBMOs that
Combining these two rank equalities leads to (2.41). Also by (2.39) and (2.40),
Combining these two equalities leads to (2.42).□
We can finally prove a group of fundamental range equalities associated with the rank equalities in (2.1)–(2.6).
Theorem 2.9
Let
Proof
We readily see from the definition of the range of matrix that
Also by (2.39), we can rewrite (2.1)–(2.6) as
Applying Theorem 2.8 to the above range and dimension results leads to (2.43)–(2.48).□
The preceding results show that it is always possible to establish nontrivial rank formulas by means of constructing certain specified block matrices and making two kinds of calculations of the ranks of the block matrices by EBMOs. The author will present more specified rank equalities and their consequences in Sections 4 and 5 using the BMM.
3 Miscellaneous algebraic identities associated with two and three idempotent matrices
In this section, the author gathers a wide range of known or novel algebraic identities that are composed by the conventional matrix operations of two and three idempotent matrices and gives a variety of meaningful and interesting consequences and applications of these matrix identities. In the following, the notation
is used for two square matrices of the same sizes. The author starts with a list of equivalent facts for a matrix to be idempotent, which were summarized by Trenkler [42,43] as follows:
Lemma 3.1
[42,43] Let
A is an idempotent.
PAQ is idempotent for any matrices
A admits the decomposition
The above lemma gives a clearer picture that the idempotency of a square matrix can be characterized in terms of other conventional matrix operations. On the other hand, there are a lot of opportunities to encounter idempotent matrices in different branches of matrix theory and applications. Here, the author refers a few of the examples that display the practical appearance of idempotent matrices:
Idempotent matrices occur widely in the theory of generalized inverses of matrices, such as the two products
Idempotent matrices play an important role in the theory of linear statistical models, in particular, the idempotent matrix
If
Any square matrix A that satisfies a quadratic equation
These apparent facts show that idempotency is one of the fundamental and intrinsic properties associated with matrices and their operations and thus can be used to describe algebraic performances of matrices under various assumptions. Because the idempotency of a matrix can be used to simplify multiplication operations of matrices, people can formulate various algebraic identities that involve idempotent matrices and use these identities in the investigation of various specified matrix analysis problems.
Next, the author presents an integrated account of algebraic matrix identities that are composed by two and three idempotent matrices and gives a variety of consequences and applications of these identities in determining the nonsingularity and the standard inverses of the matrices involved. The following three theorems are fundamental for establishing different kinds of matrix identities for two given idempotent matrices of the same order and their algebraic operations.
Theorem 3.2
[5] Given two idempotent matrices
hold, where
Proof
Equation (3.1) can be verified by multiplying the matrices on the right-hand sides of the equalities. (3.2) is a direct consequence of (3.1) by the well-known reverse order law
Theorem 3.3
Given two idempotent matrices
In particular,
Proof
Equations (3.3)–(3.6) can be verified by multiplying the matrices on the right-hand sides. Equations (3.7)–(3.10) are direct consequences of (3.3)–(3.6) by reverse order laws for inverses of products of nonsingular matrices.□
Obviously, substituting (3.1) into (3.3)–(3.6), and (3.2) into (3.7)–(3.10) will yield several groups of new factorization equalities and new reverse order laws.
Theorem 3.4
[44] Given two idempotent matrices
where
Proof
Multiplying the matrices on the right-hand sides of (3.11) yields the two equalities. (3.12) follows directly from (3.11).□
Equations (3.1)–(3.12) show that there exist essential links among any two given idempotent matrices of the same size. There is no doubt that (3.1)–(3.12) can be used to describe algebraic performances of the matrix expressions on the left-hand sides of the equalities, such as the ranks, ranges, nullity, r-potency, nilpotency, nonsingularity, inverses, generalized inverses, and norms of these matrix expressions. Note, in particular, that the term
Theorem 3.5
Given two idempotent matrices
The matrix identities below hold
(3.13)(3.14)(3.15)(3.16)(3.17)(3.18)The matrix identities below hold
(3.19)(3.20)(3.21)(3.22)(3.23)(3.24)The matrix identities below hold
(3.25)(3.26)(3.27)(3.28)- (3.29)(3.30)(3.31)(3.32)(3.33)(3.34)
- (3.35)(3.36)(3.37)(3.38)(3.39)(3.40)
The following matrix identities hold
Proof
Equations (3.13)–(3.24) follow from (3.25)–(3.28) for
Theorem 3.6
Given two idempotent matrices
and the following matrix identities hold
Proof
Equations (3.47)–(3.50) follow directly from the facts
In addition, it is easy to prove the following identities for two idempotent matrices of the same size.
Theorem 3.7
Given two idempotent matrices
The matrix identities below hold
(3.71)(3.72)(3.73)(3.74)(3.75)(3.76)(3.77)(3.78)(3.79)(3.80)(3.81)(3.82)The matrix identities below hold
In the remaining part of this section, the author constructs several general matrix identities composed by three idempotent matrices and their algebraic operations.
Theorem 3.8
Given three idempotent matrices
The matrix identities below hold
(3.87)(3.88)(3.89)(3.90)(3.91)The matrix identities below hold
(3.92)(3.93)(3.94)(3.95)The equivalent facts below hold
The equivalent facts below hold
Proof
Multiplying the matrices on the right-hand sides of (3.87), (3.88), and (3.91) yields the three equalities. The sum and difference of (3.87) and (3.88) result in (3.89) and (3.90), respectively. Equations (3.92)–(3.95) follow from direct expansions. Results (c)–(h) follow from (3.87)–(3.95).□
Apparently, the algebraic matrix identities in Theorem 3.8 can be extended without much effort to a family of idempotent matrices of the same size. This fact demonstrates that all idempotent matrices are undoubtedly linked to each other through a variety of nontrivial matrix equalities, which in turn can be utilized to describe algebraic performance of matrix expressions that involve idempotent matrices.
4 Miscellaneous rank and range formulas for idempotent matrices
People have posed and approached a large number of problems on idempotent matrices, one of which is to establish various exact formulas for calculating the ranks of matrix expressions that are composed by idempotent matrices. With the background material we have accumulated in the previous sections, it is possible to establish a wide range of analytical formulas for calculating the ranks of matrix expressions composed by idempotent matrices and obtain many meaningful consequences, including the characterizations of relationships among given idempotent matrices and their operations and the derivation of matrix range equalities.
The author begins with a group of fundamental rank equalities for two idempotent matrices and their consequences that arise from Lemma 2.1.
Theorem 4.1
Given two idempotent matrices
The rank equalities below hold
(4.1)(4.2)(4.3)(4.4)(4.5)(4.6)(4.7)(4.8)and the rank inequalities below hold
(4.9)(4.10)(4.11)(4.12)
Proof
Choosing
Theorem 4.2
Given two idempotent matrices
The rank equalities below hold
(4.13)(4.14)(4.15)If
If
If
Proof
Equations (4.13) and (4.14) follow from (2.18). Substituting (4.13) into (4.14) yields (4.15). Results (b)–(f) are direct consequences of (4.13)–(4.15).□
Theorem 4.3
Given two idempotent matrices
The rank equalities below hold
(4.16)(4.17)(4.18)(4.19)The equivalent facts below hold
(4.20)(4.21)(4.22)(4.23)The equivalent facts below hold
Proof
Note from Theorem 3.1(m) that
establishing (4.16). Equations (4.17)–(4.19) can be established by a similar approach. Setting both sides of (4.16)–(4.19) equal to zero leads to the equivalent facts in (4.20)–(4.23). Setting both sides of (4.17)–(4.19) equal to
Theorem 4.4
[36] Given two idempotent matrices
The rank equalities below hold
(4.26)(4.27)If
If
Proof
Setting
Theorem 4.5
Given two idempotent matrices
The rank equalities below hold
(4.28)(4.29)If
Proof
Applying (2.7) to the two pairs of products
Theorem 4.6
Given two idempotent matrices
The rank equalities below hold
(4.30)(4.31)(4.32)
Proof
Setting
Theorem 4.7
Given two idempotent matrices
The rank equalities below hold
(4.33)(4.34)(4.35)(4.36)(4.37)(4.38)(4.39)If
If
Proof
Result (a) follows directly from (3.1). It can be deduced by EBMOs that
On the other hand, we obtain from
Combining these two rank equalities yields (4.33). Applying (2.11)–(2.13) to the two block matrices in (4.33) yields the equalities in (4.34) and (4.35), respectively. By (2.14) and EBMOs,
Combining these equalities leads to (4.36) and (4.37). By (2.13) and EBMOs,
and
establishing (4.38) and (4.39). Results (b)–(f) are direct consequences of (4.33)–(4.39).
By (4.35), the two inequalities
Theorem 4.8
Given two idempotent matrices
The rank equalities below hold
(4.40)(4.41)(4.42)(4.43)(4.44)(4.45)(4.46)(4.47)(4.48)
Proof
as required in (4.40). Equation (4.41) can be established similarly. Next, by (4.33) and (2.13),
and by (2.14) and EBMO,
Combining these equalities leads to the equalities in (4.42). Applying (4.37) to
establishing (4.47). Equation (4.48) can be established similarly. Results (b)–(m) are direct consequences of (4.40)–(4.48).□
More rank formulas for two idempotent matrices and miscellaneous consequences are given below.
Theorem 4.9
Given two idempotent matrices
Proof
Follows from verifications by EBMOs.□
Theorem 4.10
Given two idempotent matrices
The rank formulas below hold
(4.53)(4.54)(4.55)(4.56)(4.57)(4.58)(4.59)(4.60)(4.61)(4.62)(4.63)(4.64)(4.65)(4.66)(4.67)(4.68)(4.69)(4.70)(4.71)(4.72)(4.73)(4.74)(4.75)(4.76)(4.77)(4.78)(4.79)(4.80)The range formulas below hold
(4.81)(4.82)(4.83)(4.84)(4.85)(4.86)(4.87)(4.88)(4.89)(4.90)(4.91)(4.92)(4.93)(4.94)(4.95)(4.96)(4.97)(4.98)(4.99)(4.100)(4.101)(4.102)(4.103)(4.104)(4.105)(4.106)(4.107)(4.108)The equivalent facts below hold
〈1〉
〈2〉
〈3〉
〈4〉
〈5〉
〈6〉
〈7〉
〈8〉
〈9〉
〈10〉
〈11〉
〈12〉
〈13〉
〈14〉
〈15〉
〈16〉
〈17〉
〈18〉
〈19〉
〈20〉
〈21〉
〈22〉
〈23〉
〈24〉
〈25〉
〈26〉
〈27〉
〈28〉
〈29〉
〈30〉
The equivalent facts below hold
〈1〉
〈2〉
〈3〉
〈4〉
〈5〉
〈6〉
〈7〉
〈8〉
〈9〉
〈10〉
〈11〉
〈12〉
〈13〉
〈14〉
〈15〉
〈16〉
〈17〉
〈18〉
〈19〉
〈20〉
The equivalent facts below hold
〈1〉
〈2〉
〈3〉
〈4〉
〈5〉
〈6〉
〈7〉
〈8〉
〈9〉
〈10〉
〈11〉
〈12〉
〈13〉
〈14〉
〈15〉
〈16〉
〈17〉
〈18〉
Proof
It can be deduced from (3.13), (3.59), and EBMOs that
Combining these rank equalities leads to (4.53). Equations (4.54)–(4.60), (4.62)–(4.64), (4.66)–(4.72) and (4.74)–(4.76) follow from applying (2.1)–(2.3) to the left-hand sides of the equalities. Equations (4.61) and (4.73) follow from applying (2.1) to (3.59). Equation (4.65) follows from applying (2.1) and (4.61) to (3.14). Equation (4.77) follows from applying (2.6) and (3.55) to (3.20). By (3.13), (3.59), and block elementary operations of matrices,
Combining these rank equalities with (3.60), (3.65), (4.61), and (4.73) leads to (4.78). Equations (4.79)–(4.84) follow from applying (2.1) to (3.66)–(3.70). Results (b) follow from applying (2.43)–(2.48) and (4.49)–(4.52) to (4.53)–(4.84). Results (c) and (d) are direct consequences of (4.53)–(4.84).□
Theorem 4.11
Given two idempotent matrices
where
Proof
Equation (4.109) follows from (3.11). Equation (4.110) follows from (4.109) by setting
Theorem 4.12
Given two idempotent matrices
The rank equalities below hold
(4.111)(4.112)(4.113)(4.114)
Proof
Applying (2.1) and (4.42) to
Theorem 4.13
Given two idempotent matrices
The rank equalities below hold
(4.115)(4.116)(4.117)(4.118)(4.119)(4.120)(4.121)(4.122)(4.123)(4.124)
Proof
Let
Combining these two rank equalities leads to (4.123). Further by
Combining these two rank equalities leads to (4.124). Results (b)–(g) follow directly from (4.115), (4.116), and (4.119)–(4.122). By (4.123), the two inequalities
We next derive from Theorem 3.8 some rank and range formulas for expressions composed by three idempotent matrices and their consequences.
Theorem 4.14
Given three idempotent matrices
The following rank formulas hold
(4.125)(4.126)(4.127)for
The following range equalities
(4.128)(4.129)(4.130)hold for
Proof
Applying (2.1) to (3.92)–(3.94) leads to (4.125)–(4.127). Results (b)–(e) follow from (4.125)–(4.127).Furthermore, it is desired to deduce various expansion formulas for calculating the ranks of matrix expressions composed by a family of idempotent matrices. Next, the author presents some general rank equalities associated with a family of idempotent matrices.□
Theorem 4.15
Given a family of idempotent matrices
The rank equalities below hold
(4.131)(4.132)(4.133)(4.134)
establishing (4.132), this proof was first given in [39]. Replacing
establishing (4.133). Replacing
Theorem 4.16
Given a family of idempotent matrices
The rank equality below holds
(4.135)where
If
Proof
A much trickier block matrix composed by the family of idempotent matrices is given by
It is easy to verify that the above construction leads to the following two rank equalities:
Combining these two equalities leads to (4.135). Results (b)–(f) follow directly from (4.135).□
It is easy to see that (4.135) for
Corollary 4.17
Given three idempotent matrices
The rank equality below holds
If
If
Finally, we deduce from Theorem 4.13(a) the following result on the ranks of a partitioned matrix and generalized inverses of its submatrices.
Corollary 4.18
Given
holds for all
The maximum and minimum ranks of (4.136) with respect to
(4.137)(4.138)
Proof
By definition,
holds for all
as required for (4.136). Applying the following two known rank formulas
in [30] to
Substituting these two results into (4.140) yields
Combining (4.136) with (4.141) and (4.142) leads to (4.137) and (4.138), respectively. Setting the both sides of (4.137) and (4.138) equal to zero leads to Results (b) and (c).□
5 Bounds of ranks of some matrix pencils composed by two idempotent matrices
Since the rank of a matrix expression is a function of the variable entries in the expression, which may vary with respect to the different choices of the entries, people wish to know the exact upper and lower bounds of the ranks (maximum and minimum ranks) of the matrix expression under various assumptions. By the definitions of generalized inverses, both
associated with the idempotent matrices
Lemma 5.1
[30] The maximum and minimum ranks of the linear matrix-valued function
where
Lemma 5.2
[6,10,25] Let
where
Theorem 5.3
Let
If
(5.5)(5.6)In particular, the following results hold.
There always exist
There do not exist
The following two formulas hold
(5.7)(5.8)In particular, the following results hold.
There exist
There exist
The rank of
The following two formulas hold
(5.9)(5.10)In particular, the following results hold.
There exist
There exist
The rank of
The following two formulas hold
(5.11)(5.12)In particular, the following results hold.
There exist
There exist
The rank of
Proof
By (5.4),
where
and
Substituting different values of
Theorem 5.4
Let
If
(5.16)(5.17)In particular, the following results hold.
There always exist
There do not exist
The following two formulas hold
(5.18)(5.19)In particular, the following results hold.
There exist
There exist
The rank of
The following two formulas hold
(5.20)(5.21)In particular, the following results hold.
There exist
There exist
The rank of
Proof
By (5.4),
where
and
Substituting different values of
Theorem 5.5
Let
If
In particular, the following results hold.
There always exist
There do not exist
The following two formulas hold
In particular, the following results hold.
There exist
There exist
The rank of
The following two formulas hold
In particular, the following results hold.
There exist
There exist
The rank of
The following two formulas hold
In particular, the following results hold.
There exist
There exist
The rank of
Proof
By (5.4),
where
and
Substituting different values of λ into the above formulas and simplifying yield the rank formulas required.□
Theorem 5.6
Let
If
In particular, the following results hold.
There always exist
There do not exist
The following two formulas hold
In particular, the following results hold.
There exist
There exist
The rank of
The following two formulas hold
In particular, the following results hold.
There exist
There exist
The rank of
Proof
By (5.4),
where
and
Substituting different values of
Setting
Corollary 5.7
Let
If
In particular, the following results hold.
There always exists an
There does not exist an
The following two formulas hold:
In particular, the following results hold.
There exists an
There exists an
The rank of
The following two formulas hold
In particular, the following results hold.
There exists an
There exists an
The rank of
The following two formulas hold
In particular, the following results hold.
There exists an
There exists an
The rank of
Corollary 5.8
Let
If
In particular, the following results hold.
There always exists an
There does not exist an
The following two formulas hold
In particular, the following results hold.
There exists an
There exists an
The rank of
The following two formulas hold
In particular, the following results hold.
There exists an
There exists an
The rank of
Finally, the author presents a group of rank formulas associated with the products of two block matrices and their generalized inverses.
Corollary 5.9
Let
In particular, the following formulas hold.
Setting the both sides of (5.26)–(5.30) equal to
6 Concluding remarks
The author has presented a large number of known and novel algebraic identities that are composed by idempotent matrices and established many explicit formulas for calculating the ranks and ranges of idempotent matrices by means of the BMM and EBMOs. The author has also presented a variety of applications of these identities and formulas in dealing with a variety of issues associated with idempotent matrices under various assumptions. Note that all the preceding results and facts are presented in elementary and analytical forms that fall in the scope of elementary linear algebra and matrix theory, so that the work is accepted easily for all readers with a background of college mathematics, and thus, the author hopes that they can be used in the teaching and research of idempotent matrix problems and also hopes that part of the contents in this survey paper can be chosen as constructive materials in the compilation of new textbooks and handbooks on linear algebra and matrix mathematics. Furthermore, this work shows that the equalities and inequalities of matrices have been one of the most attractive and fruitful research topics in linear algebra and also demonstrates that there still exist many simple and interesting problems on fundamental objects in linear algebra for which we can make deeper exploration and find out various novel and intrinsic conclusions.
It is expected that more algebraic identities composed by idempotent matrices and more analytical formulas for calculating the ranks of matrix expressions can be established by using various tricky block matrix constructions, which we believe will bring a great increase of classic knowledge in linear algebra and will also provide useful tools to deal with various challenging problems in matrix analysis and applications. As a continuation of this research, the author proposes some problems for consideration:
Establish analytical formulas for calculating the ranks of the following general matrix expressions:
with respect to the multiple generalized inverses in them.
and
where
On the other hand, it is well known that idempotents and generalized inverses of elements can symbolically be defined in rings and operator algebras, which have been recognized as important tools in the investigation of issues in these disciplines. Much to his regret, the author has nothing to say in this paper about idempotents and generalized inverses of elements in general algebraic structures. Nevertheless, it would be very attractive to be able to generalize by algebraic and deductive calculations of the preceding formulas, results, and facts to the problems that are somehow close to the complex matrix case, as they also encode more interesting issues on idempotents in general algebraic structures.
Acknowledgments
The author is grateful to an anonymous referee for his/her helpful comments and suggestions on this paper.
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- Quasilinear Dirichlet problems with competing operators and convection
- Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
- Special Issue on Evolution Equations, Theory and Applications
- Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
- Three classes of decomposable distributions
