Pythagorean fuzzy $ N $-Soft PROMETHEE approach: A new framework for group decision making

Muhammad Akram; Maheen Sultan; Arooj Adeel; Mohammed M. Ali Al-Shamiri; Muhammad Akram; Maheen Sultan; Arooj Adeel; Mohammed M. Ali Al-Shamiri

doi:10.3934/math.2023887

AIMS Mathematics

2023, Volume 8, Issue 8: 17354-17380. doi: 10.3934/math.2023887

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Research article Special Issues

Pythagorean fuzzy $N$ -Soft PROMETHEE approach: A new framework for group decision making

1.
Department of Mathematics, University of the Punjab, New Campus, Lahore- 54590, Pakistan
2.
Department of Mathematics, Division of Science and Technology, University of Education, Lahore-54770, Pakistan
3.
Department of Mathematics, Faculty of science and arts, Mahayl Assir, King Khalid University, Saudi Arabia

Received: 08 March 2023 Revised: 30 April 2023 Accepted: 05 May 2023 Published: 19 May 2023
MSC : 03E72, 03E75, 90B50

The use of Pythagorean fuzzy $N$ -soft sets (PFNSs) enables the examination of belongingness and non-belongingness of membership degrees, as well as their combinations with $N$ -grading, in the unpredictable nature of individuals. This research aims to enhance our understanding of a popular multi-criteria group decision making (MCGDM) technique, Preference Ranking Organization Method for Enrichment of Evaluations, under the PFNS environment, aiding in making effective decisions for real-life problems, as fuzzy set theory is directly relevant to real-life applications. The PROMETHEE technique's main principle is to calculate the inflow and outflow streams of alternatives based on the deviation of their score degrees, ultimately providing partial and complete rankings of the given options. To capture the uncertainty of human nature, which demands both the association and disassociation of the considered criteria and provision of $N$ -grading, the PFNS PROMETHEE technique is introduced in this research article. First, an Analytic Hierarchy Process AHP is used to check the feasibility of the standard weights of the criteria. The article then explains the detailed method of the fuzzy $N$ -soft PROMETHEE technique to rank alternatives, with all the steps presented in an extensive flowchart for better understanding of the methodology. Furthermore, the practicality and viability of the proposed technique are demonstrated through an example of selecting the best chemical element in cloud seeding, where the most suitable choice is identified using an outranking directed graph. The credibility of the PFNS PROMETHEE technique is assessed by comparison with an existing method. Finally, the proposed technique's strengths and weaknesses are discussed to demonstrate its efficiency and drawbacks.

Keywords:

Citation: Muhammad Akram, Maheen Sultan, Arooj Adeel, Mohammed M. Ali Al-Shamiri. Pythagorean fuzzy $N$ -Soft PROMETHEE approach: A new framework for group decision making[J]. AIMS Mathematics, 2023, 8(8): 17354-17380. doi: 10.3934/math.2023887

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Abstract

1. Introduction

Throughout this article, ${\mathbb C}^{m \times n}$ denotes the collection of all $m \times n$ matrices over the field of complex numbers, $A^{\ast}$ denotes the conjugate transpose of $A \in {\mathbb C}^{m \times n}$ , $r(A)$ denotes the rank of $A \in {\mathbb C}^{m \times n}$ , ${\mathscr R}(A)$ denotes the range of $A \in {\mathbb C}^{m \times n}$ , $I_m$ denotes the identity matrix of order $m$ , and $[\, A, \, B\, ]$ denotes a row block matrix consisting of $A\in {\mathbb C}^{m \times n}$ and $B\in {\mathbb C}^{m \times p}$ . The Moore-Penrose generalized inverse of $A \in {\mathbb C}^{m \times n}$ , denoted by $A^{\dagger}$ , is the unique matrix $X \in {\mathbb C}^{n \times m}$ satisfying the four Penrose equations

${\rm (1)} \ AXA = A, \ \ {\rm (2)} \ XAX = X, \ \ {\rm (3)} \ (AX)^{*} = AX, \ \ {\rm (4)} \ (XA)^{*} = XA.$

Further, we denote by

$\begin{align} P_A = AA^{\dagger}, \ \ E_{A} = I_m - AA^{\dagger} \end{align}$

(1.1)

the two orthogonal projectors induced from $A\in {\mathbb C}^{m \times n}$ . For more detailed information regarding generalized inverses of matrices, we refer the reader to ^[2,3,4].

Recall that the well-known Kronecker product of any two matrices $A = (a_{ij}) \in {\mathbb C}^{m \times n}$ and $B = (b_{ij}) \in {\mathbb C}^{p \times q}$ is defined to be

$A \otimes B = (a_{ij}B) = \begin{bmatrix} a_{11}B & a_{12}B & \cdots & a_{1n}B \\ a_{21}B & a_{22}B & \cdots & a_{2n}B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}B & a_{m2}B & \cdots & a_{mn}B \end{bmatrix} \in {\mathbb C}^{mp \times nq}.$

The Kronecker product, named after German mathematician Leopold Kronecker, was classified to be a special kind of matrix operation, which has been regarded as an important matrix operation and mathematical technique. This product has wide applications in system theory, matrix calculus, matrix equations, system identification and more (cf. ^{[1,5,6,7,8,9,10,11,12,13,14,16,18,19,20,21,23,24,25,27,28,33,34]}). It has been known that the matrices operations based on Kronecker products have a series of rich and good structures and properties, and thus they have many significant applications in the research areas of both theoretical and applied mathematics. In fact, mathematicians established a variety of useful formulas and facts related to the products and used them to deal with various concrete scientific computing problems. Specifically, the basic facts on Kronecker products of matrices in the following lemma were highly appraised and recognized (cf. ^{[15,17,21,32,34]}).

Fact 1.1. Let $A \in {\mathbb C}^{m \times n},$ $B \in {\mathbb C}^{p \times q},$ $C \in {\mathbb C}^{n \times s},$ and $D\in {\mathbb C}^{q \times t}.$ Then $,$ the following equalities hold $:$

$\begin{align} & \ \ \ \ \ \ (A \otimes B)(C \otimes D) = (AC) \otimes (BD), \end{align}$

(1.2)

$\begin{align} & (A \otimes B)^{*} = A^{*} \otimes B^{*}, \ \ \ (A \otimes B)^{\dagger} = A^{\dagger} \otimes B^{\dagger}, \end{align}$

(1.3)

$\begin{align} & P_{A\otimes B} = P_{A} \otimes P_{B}, \ \ \ \ \ \ r(A \otimes B) = r(A)r(B). \end{align}$

(1.4)

In addition, the Kronecker product of matrices has a rich variety of algebraic operation properties. For example, one of the most important features is that the product $A_1 \otimes A_2$ can be factorized as certain ordinary products of matrices:

$\begin{align} A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{n_1} \otimes A_2) = (I_{m_1} \otimes A_2)(A_1 \otimes I_{n_2}) \end{align}$

(1.5)

for any $A_1 \in {\mathbb C}^{m_1 \times n_1}$ and $A_2 \in {\mathbb C}^{m_2 \times n_2}$ , and the triple Kronecker product $A_1 \otimes A_2 \otimes A_3$ can be written as

$\begin{align} A_1 \otimes A_2 \otimes A_3 & = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{n_1} \otimes A_2 \otimes I_{m_3})(I_{n_1} \otimes I_{n_2} \otimes A_3) , \end{align}$

(1.6)

$\begin{align} A_1 \otimes A_2 \otimes A_3 & = (I_{n_1} \otimes A_2 \otimes I_{m_3})(I_{n_1}(I_{n_1} \otimes I_{n_2} \otimes A_3) (A_1\otimes I_{m_2} \otimes I_{m_3}), \end{align}$

(1.7)

$\begin{align} A_1 \otimes A_2 \otimes A_3 & = (I_{m_1} \otimes I_{m_2} \otimes A_3)(A_1 \otimes I_{n_2} \otimes I_{n_3})(I_{m_1} \otimes A_2 \otimes I_{n_3}) \end{align}$

(1.8)

for any $A_1 \in {\mathbb C}^{m_1 \times n_1}$ , $A_2 \in {\mathbb C}^{m_2 \times n_2}$ and $A_3 \in {\mathbb C}^{m_3 \times n_3}$ , where the five matrices in the parentheses on the right hand sides of (1.5)–(1.8) are usually called the dilation expressions of the given three matrices $A_1$ , $A_2$ and $A_3$ , and the four equalities in (1.5)–(1.8) are called the dilation factorizations of the Kronecker products $A_1 \otimes A_2$ and $A_1 \otimes A_2 \otimes A_3$ , respectively. A common feature of the four matrix equalities in (1.5)–(1.8) is that they factorize Kronecker products of any two or three matrices into certain ordinary products of the dilation expressions of $A_1$ , $A_2$ and $A_3$ . Particularly, a noticeable fact we have to point out is that the nine dilation expressions of matrices in (1.5)–(1.8) commute each other by the well-known mixed-product property in (1.2) when $A_1$ , $A_2$ and $A_3$ are all square matrices. It can further be imagined that there exists proper extension of the dilation factorizations to Kronecker products of multiple matrices. Although the dilation factorizations in (1.5)–(1.8) seem to be technically trivial in form, they can help deal with theoretical and computational issues regarding Kronecker products of two or three matrices through the ordinary addition and multiplication operations of matrices.

In this article, we provide a new analysis of performances and properties of Kronecker products of matrices, as well as present a wide range of novel and explicit facts and results through the dilation factorizations described in (1.5)–(1.8) for the purpose of obtaining a deeper understanding and grasping of Kronecker products of matrices, including a number of analytical formulas for calculating ranks, dimensions, orthogonal projectors, and ranges of the dilation expressions of matrices and their algebraic operations.

The remainder of article is organized as follows. In section two, we introduce some preliminary facts and results concerning ranks, ranges, and generalized inverses of matrices. In section three, we propose and prove a collection of useful equalities, inequalities, and formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges associated with the Kronecker products $A_1 \otimes A_2$ and $A_1 \otimes A_2 \otimes A_3$ through the dilation expressions of $A_1$ , $A_2$ and $A_3$ and their operations. Conclusions and remarks are given in section four.

2. Some preliminaries

One of the remarkable applications of generalized inverses of matrices is to establish various exact and analytical expansion formulas for calculating the ranks of partitioned matrices. As convenient and skillful tools, these matrix rank formulas can be used to deal with a wide variety of theoretical and computational issues in matrix theory and its applications (cf. ^[22]). In this section, we present a mixture of commonly used formulas and facts in relation to ranks of matrices and their consequences about the commutativity of two orthogonal projectors, which we shall use as analytical tools to approach miscellaneous formulas related to Kronecker products of matrices.

Lemma 2.1. ^[22,29] Let $A \in {\mathbb C}^{m \times n}$ and $B \in {\mathbb C}^{ m \times k}.$ Then $,$ the following rank equalities

$\begin{align} & r[A, \, B] = r(A) + r(E_AB) = r(B) + r(E_BA), \end{align}$

(2.1)

$\begin{align} & r[A, \, B] = r(A) + r(B) - 2r(A^*B) + r[P_{A}P_{B}, \, P_{B}P_{A}], \end{align}$

(2.2)

$\begin{align} & r[A, \, B] = r(A) + r(B) - r(A^*B) + 2^{-1}r(P_AP_B - P_BP_A), \end{align}$

(2.3)

$\begin{align} & r[A, \, B] = r(A) + r(B) - r(A^*B) + r(P_{[A, \, B]} - P_A - P_B + P_AP_B) \end{align}$

(2.4)

hold $.$ Therefore $,$

$\begin{align} P_AP_B = P_BP_A &\Leftrightarrow P_{[A, \, B]} = P_A + P_B - P_AP_B \\ &\Leftrightarrow r(E_AB) = r(B) - r(A^*B) \\ & \Leftrightarrow r[A, \, B] = r(A) + r(B) - r(A^*B) \\ & \Leftrightarrow {\mathscr R}(P_{A}P_{B}) = {\mathscr R}(P_{B}P_{A}). \end{align}$

(2.5)

If $P_AP_B = P_BP_A,$ $P_AP_C = P_CP_A$ and $P_BP_C = P_CP_B,$ then

$\begin{align} P_{[A, \, B, \, C]} = P_A + P_B + P_C - P_AP_B - P_AP_C - P_BP_C + P_AP_BP_C. \end{align}$

(2.6)

Lemma 2.2. ^[30] Let $A, \, B$ and $C \in {\mathbb C}^{m \times m}$ be three idempotent matrices $.$ Then $,$ the following rank equality

$\begin{align} r[A, \, B, \, C] & = r(A) + r(B) + r(C) - r[AB, \, AC] - r[BA, \, BC] - r[CA, \, CB] \\ & \ \ \ + r[AB, \, AC, \, BA, \, BC, \, CA, \, CB] \end{align}$

(2.7)

holds $.$ As a special instance $,$ if $AB = BA,$ $AC = CA$ and $BC = CB,$ then

$\begin{align} r[A, \, B, \, C] & = r(A) + r(B) + r(C) - r[AB, \, AC] - r[BA, \, BC] - r[CA, \, CB] + r[AB, \, AC, \, BC]. \end{align}$

(2.8)

The formulas and facts in the above two lemmas belong to mathematical competencies and conceptions in ordinary linear algebra. Thus they can easily be understood and technically be utilized to establish and simplify matrix expressions and equalities consisting of matrices and their generalized inverses.

3. Main results

We first establish a group of formulas and facts associated with the orthogonal projectors, ranks, dimensions, and ranges of the matrix product in (1.5).

Theorem 3.1. Let $A_1 \in {\mathbb C}^{m_1 \times n_1}$ and $A_2 \in {\mathbb C}^{m_2 \times n_2},$ and denote by

$M_1 = A_1 \otimes I_{m_2}, \ \ M_2 = I_{m_1} \otimes A_2$

the two dilation expressions of $A_1$ and $A_2,$ respectively $.$ Then $,$ we have the following results $.$

${\rm (a)}$ The following orthogonal projector equalities hold $:$

$\begin{align} P_{A_1 \otimes A_2} = P_{M_1}P_{M_2} = P_{M_2}P_{M_1} = P_{A_1} \otimes P_{A_2}. \end{align}$

(3.1)

${\rm (b)}$ The following four rank equalities hold $:$

$\begin{align} r[A_1 \otimes I_{m_2}, \, I_{m_1} \otimes A_2] & = m_1m_2 - (m_1 - r(A_1))(m_2 - r(A_2)), \end{align}$

(3.2)

$\begin{align} r[A_1 \otimes I_{m_2}, \, I_{m_1} \otimes E_{A_2}] & = m_1m_2 - (m_1 - r(A_1))r(A_2), \end{align}$

(3.3)

$\begin{align} r[E_{A_1} \otimes I_{m_2}, \, I_{m_1} \otimes A_2] & = m_1m_2 - r(A_1)(m_2 - r(A_2)), \end{align}$

(3.4)

$\begin{align} r[E_{A_1} \otimes I_{m_2}, \, I_{m_1} \otimes E_{A_2}] & = m_1m_2 - r(A_1)r(A_2), \end{align}$

(3.5)

and the following five dimension equalities hold $:$

$\begin{align} \dim( {\mathscr R}(M_1)\cap {\mathscr R}(M_2)) & = r(M_1M_2) = r(A_1)r(A_2), \end{align}$

(3.6)

$\begin{align} \dim( {\mathscr R}(M_1)\cap {\mathscr R}^{\perp}(M_2)) & = r(M_1E_{M_2}) = r(A_1)(m_2 - r(A_2)), \end{align}$

(3.7)

$\begin{align} \dim( {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}(M_2)) & = r(E_{M_1}M_2) = (m_1 - r(A_1))r(A_2), \end{align}$

(3.8)

$\begin{align} \dim( {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}^{\perp}(M_2)) & = r(E_{M_1}E_{M_2}) = (m_1 - r(A_1))(m_2 - r(A_2)), \end{align}$

(3.9)

$\begin{align} &\dim( {\mathscr R}(M_1)\cap {\mathscr R}(M_2)) + \dim( {\mathscr R}(M_1)\cap {\mathscr R}^{\perp}(M_2)) + \dim( {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}^{\perp}(M_2)) \\ & \ + \dim( {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}^{\perp}(M_2)) = m_1m_2. \end{align}$

(3.10)

${\rm (c)}$ The following range equalities hold $:$

$\begin{align} {\mathscr R}(M_1)\cap {\mathscr R}(M_2) & = {\mathscr R}(M_1M_2) = {\mathscr R}(M_2M_1) = {\mathscr R}(A_1 \otimes A_2), \end{align}$

(3.11)

$\begin{align} {\mathscr R}(M_1)\cap {\mathscr R}^{\perp}(M_2) & = {\mathscr R}(M_1E_{M_2}) = {\mathscr R}(E_{M_2}M_1) = {\mathscr R}(A_1 \otimes E_{A_2}), \end{align}$

(3.12)

$\begin{align} {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}(M_2) & = {\mathscr R}(E_{M_1}M_2) = {\mathscr R}(M_2E_{M_1}) = {\mathscr R}(E_{A_1} \otimes A_2), \end{align}$

(3.13)

$\begin{align} {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}^{\perp}(M_2) & = {\mathscr R}(E_{M_1}E_{M_2}) = {\mathscr R}(E_{M_2}E_{M_1}) = {\mathscr R}(E_{A_1} \otimes E_{A_2}), \end{align}$

(3.14)

$\begin{align} & ( {\mathscr R}(M_1)\cap {\mathscr R}(M_2)) \oplus ( {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}(M_2)) \oplus ( {\mathscr R}(M_1)\cap {\mathscr R}^{\perp}(M_2)) \oplus ( {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}^{\perp}(M_2)) \\ & = {\mathbb C}^{m_1m_2}. \end{align}$

(3.15)

${\rm (d)}$ The following orthogonal projector equalities hold $:$

$\begin{align} &P_{ {\mathscr R}(M_1)\cap {\mathscr R}(M_2)} = P_{M_1}P_{M_2} = P_{A_1}\otimes P_{A_2}, \end{align}$

(3.16)

$\begin{align} & P_{ {\mathscr R}(M_1)\cap {\mathscr R}^{\perp}(M_2)} = P_{M_1}E_{M_2} = P_{A_1}\otimes E_{A_2}, \end{align}$

(3.17)

$\begin{align} &P_{ {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}(M_2)} = E_{M_1}P_{M_2} = E_{A_1}\otimes P_{A_2}, \end{align}$

(3.18)

$\begin{align} &P_{ {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}^{\perp}(M_2)} = E_{M_1}E_{M_2} = E_{A_1}\otimes E_{A_2}, \end{align}$

(3.19)

$\begin{align} & P_{ {\mathscr R}(M_1)\cap {\mathscr R}(M_2)} + P_{ {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}(M_2)} + P_{ {\mathscr R}(M_1)\cap {\mathscr R}^{\perp}(M_2)} + P_{ {\mathscr R}^{\perp}(M_1)\cap {\mathscr R}^{\perp}(M_2)} = I_{m_1m_2}. \end{align}$

(3.20)

${\rm (e)}$ The following orthogonal projector equalities hold $:$

$\begin{align} P_{[A_1 \otimes I_{m_2}, \, I_{m_1} \otimes A_2]} & = P_{A_1} \otimes I_{m_2} + I_{m_1} \otimes P_{A_2} - P_{A_1} \otimes P_{A_2} = I_{m_1m_2} - E_{A_1} \otimes E_{A_2}, \end{align}$

(3.21)

$\begin{align} P_{[A_1 \otimes I_{m_2}, \, I_{m_1} \otimes E_{A_2}]} & = P_{A_1} \otimes I_{m_2} + I_{m_1} \otimes E_{A_2} - P_{A_1} \otimes E_{A_2} = I_{m_1m_2} - E_{A_1} \otimes P_{A_2}, \end{align}$

(3.22)

$\begin{align} P_{[E_{A_1} \otimes I_{m_2}, \, I_{m_1} \otimes A_2]} & = E_{A_1} \otimes I_{m_2} + I_{m_1} \otimes P_{A_2} - E_{A_1} \otimes P_{A_2} = I_{m_1m_2} - P_{A_1} \otimes E_{A_2}, \end{align}$

(3.23)

$\begin{align} P_{[E_{A_1} \otimes I_{m_2}, \, I_{m_1} \otimes E_{A_2}]} & = E_{A_1} \otimes I_{m_2} + I_{m_1} \otimes E_{A_2} - E_{A_1} \otimes E_{A_2} = I_{m_1m_2} - P_{A_1} \otimes P_{A_2}. \end{align}$

(3.24)

Proof. It can be seen from (1.2) and (1.4) that

$\begin{align*} P_{M_1}P_{M_2} & = (A_1 \otimes I_{m_2})(A_1 \otimes I_{m_2})^{\dagger}(I_{m_1} \otimes A_2)(I_{m_1} \otimes A_2)^{\dagger} \\ & = (A_1 \otimes I_{m_2})(A_1^{\dagger} \otimes I_{m_2})(I_{m_1} \otimes A_2)(I_{m_1} \otimes A_2^{\dagger}) \\ & = (P_{A_1} \otimes I_{m_2})(I_{m_1} \otimes P_{A_2}) \\ & = P_{A_1} \otimes P_{A_2}, \\ P_{M_2}P_{M_1} & = (I_{m_1} \otimes A_2)(I_{m_1} \otimes A_2)^{\dagger}(A_1 \otimes I_{m_2})(A_1 \otimes I_{m_2})^{\dagger} \\ & = (I_{m_1} \otimes A_2)(I_{m_1} \otimes A_2^{\dagger})(A_1 \otimes I_{m_2})(A_1^{\dagger} \otimes I_{m_2}) \\ & = (I_{m_1} \otimes P_{A_2})(P_{A_1} \otimes I_{m_2}) \\ & = P_{A_1} \otimes P_{A_2}, \end{align*}$

thus establishing (3.1).

Applying (2.1) to $[A_1 \otimes I_{m_2}, \, I_{m_1} \otimes A_2]$ and then simplifying by (1.2)–(1.4) yields

$\begin{align*} r[A_1 \otimes I_{m_2}, \, I_{m_1} \otimes A_2] & = r(A_1 \otimes I_{m_2}) + r((I_{m_1m_2} - (A_1 \otimes I_{m_2})(A_1 \otimes I_{m_2})^{\dagger})(I_{m_1} \otimes A_2)) \nonumber \\ & = r(A_1 \otimes I_{m_2}) + r((I_{m_1m_2} - (A_1 \otimes I_{m_2})(A_1^{\dagger} \otimes I_{m_2}))(I_{m_1} \otimes A_2)) \nonumber \\ & = r(A_1 \otimes I_{m_2}) + r((I_{m_1m_2} - (A_1A_1^{\dagger}) \otimes I_{m_2})(I_{m_1} \otimes A_2)) \nonumber \\ & = m_2r(A_1) + r((I_{m_1} - A_1A_1^{\dagger})\otimes I_{m_2})(I_{m_1} \otimes A_2)) \nonumber \\ & = m_2r(A_1) + r((I_{m_1} - A_1A_1^{\dagger})\otimes A_2)) \nonumber \\ & = m_2r(A_1) + r(I_{m_1} - A_1A_1^{\dagger})r(A_2) \nonumber \\ & = m_2r(A_1) + (m_1 - r(A_1))r(A_2) \nonumber \\ & = m_1m_2 - (m_1 - r(A_1))(m_2 - r(A_2)), \end{align*}$

as required for (3.2). In addition, (3.2) can be directly established by applying (2.5) to the left hand side of (3.2). Equations (3.3)–(3.5) can be obtained by a similar approach. Subsequently by (3.2),

$\begin{align*} & \dim( {\mathscr R}(M_1)\cap {\mathscr R}(M_2)) = r(M_1) + r(M_2) - r[M_1, \, M_2] = r(A_1)r(A_2), \end{align*}$

as required for (3.6). Equations (3.7)–(3.9) can be established by a similar approach. Adding (3.7)–(3.9) leads to (3.10).

The first two equalities in (3.11) follow from (3.6), and the last two range equalities follow from (3.1).

Equations (3.12)–(3.14) can be established by a similar approach. Adding (3.11)–(3.14) and combining with (3.10) leads to (3.15).

Equations (3.16)–(3.19) follow from (3.11)–(3.14). Adding (3.16)–(3.19) leads to (3.20).

Under (3.1), we find from (2.5) that

$P_{[M_1, \, M_2]} = P_{M_1} + P_{M_2} - P_{M_1}P_{M_2} = P_{A_1} \otimes I_{m_2} + I_{m_1} \otimes P_{A_2} - P_{A_1} \otimes P_{A_2} = I_{m_1m_2} - E_{A_1} \otimes E_{A_2},$

as required for (3.21). Equations (3.22)–(3.24) can be established by a similar approach. □

Equation (3.2) was first shown in ^[7]; see also ^[27] for some extended forms of (3.2). Obviously, Theorem 3.1 reveals many performances and properties of Kronecker products of matrices, and it is no doubt that they can be used as analysis tools to deal with various matrix equalities composed of algebraic operations of Kronecker products of matrices. For example, applying the preceding results to the Kronecker sum and difference $A_1 \otimes I_{m_2} \pm I_{m_1} \otimes A_2$ for two idempotent matrices $A_1$ and $A_2$ , we obtain the following interesting consequences.

Theorem 3.2. Let $A_1\in {\mathbb C}^{m_1 \times m_1}$ and $A_2 \in {\mathbb C}^{m_2 \times m_2}.$ Then $,$ the following rank inequality

$\begin{align} r(A_1 \otimes I_{m_2} + I_{m_1} \otimes A_2) \geq m_1r(A_2) + m_2r(A_1) - 2r(A_1)r(A_2) \end{align}$

(3.25)

holds $.$ If $A_1 = A_1^2$ and $A_2 = A_2^2,$ then the following two rank equalities hold $:$

$\begin{align} r(A_1 \otimes I_{m_2} + I_{m_1} \otimes A_2) & = m_1r(A_2) + m_2r(A_1) - r(A_1)r(A_2), \end{align}$

(3.26)

$\begin{align} r(A_1 \otimes I_{m_2} - I_{m_1} \otimes A_2) & = m_1r(A_2) + m_2r(A_1) - 2r(A_1)r(A_2). \end{align}$

(3.27)

Proof. Equation (3.25) follows from applying the following well-known rank inequality (cf. ^[22])

$\begin{align*} r(A + B) \geq r\begin{bmatrix} A \\ B \end{bmatrix} + r[A, \, B] - r(A) - r(B) \end{align*}$

and (2.1) to $A_1 \otimes I_{m_2} + I_{m_1} \otimes A_2$ . Specifically, if $A_1 = A_1^2$ and $A_2 = A_2^2,$ then it is easy to verify that $(A_1 \otimes I_{m_2})^2 = A_1^2 \otimes I_{m_2} = A_1 \otimes I_{m_2}$ and $(I_{m_1} \otimes A_2)^2 = I_{m_1} \otimes A_2^2 = I_{m_1} \otimes A_2$ under $A_1^2 = A_1$ and $A_2^2 = A_2$ . In this case, applying the following two known rank formulas

$\begin{align*} r(A + B) & = r\begin{bmatrix} A & B \\ B & 0 \end{bmatrix} - r(B) = r\begin{bmatrix} B & A \\ A & 0 \end{bmatrix} - r(A), \\ r(A - B) & = r\begin{bmatrix} A \\ B \end{bmatrix} + r[A, \, B] - r(A) - r(B), \end{align*}$

where $A$ and $B$ are two idempotent matrices of the same size (cf. ^[29,31]), to $A_1 \otimes I_{m_2} \pm I_{m_1} \otimes A_2$ and then simplifying by (2.1) and (3.2) yields (3.26) and (3.27), respectively. □

Undoubtedly, the above two theorems reveal some essential relations among the dilation forms of two matrices by Kronecker products, which demonstrate that there still exist various concrete research topics on the Kronecker product of two matrices with analytical solutions that can be proposed and obtained. As a natural and useful generalization of the preceding formulas, we next give a diverse range of results related to the three-term Kronecker products of matrices in (1.6).

Theorem 3.3. Let $A_1 \in {\mathbb C}^{m_1 \times n_1}, \, A_2 \in {\mathbb C}^{m_2 \times n_2}$ and $A_3 \in {\mathbb C}^{m_3 \times n_3},$ and let

$\begin{align} X_1 = A_1 \otimes I_{m_2} \otimes I_{m_3}, \ \ X_2 = I_{m_1} \otimes A_2 \otimes I_{m_3}, \ \ X_3 = I_{m_1} \otimes I_{m_2} \otimes A_3 \end{align}$

(3.28)

denote the three dilation expressions of $A_1,$ $A_2$ and $A_3,$ respectively $.$ Then $,$ we have the following results $.$

${\rm (a)}$ The following three orthogonal projector equalities hold $:$

$\begin{align} P_{X_1} = P_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \ \ P_{X_2} = I_{m_1} \otimes P_{A_2} \otimes I_{m_3}, \ \ P_{X_3} = I_{m_1} \otimes I_{m_2} \otimes P_{A_3}, \end{align}$

(3.29)

the following equalities hold $:$

$\begin{align} & P_{X_1}P_{X_2} = P_{X_2}P_{X_1} = P_{A_1} \otimes P_{A_2} \otimes I_{m_3}, \end{align}$

(3.30)

$\begin{align} & P_{X_1}P_{X_3} = P_{X_3}P_{X_1} = P_{A_1} \otimes I_{m_2} \otimes P_{A_3}, \end{align}$

(3.31)

$\begin{align} & P_{X_2}P_{X_3} = P_{X_3}P_{X_2} = I_{m_1} \otimes P_{A_2} \otimes P_{A_3}, \end{align}$

(3.32)

and the equalities hold $:$

$\begin{align} P_{A_1 \otimes A_2 \otimes A_3} = P_{X_1}P_{X_2}P_{X_3} = P_{A_1} \otimes P_{A_2} \otimes P_{A_3}. \end{align}$

(3.33)

${\rm (b)}$ The following eight rank equalities hold $:$

$\begin{align} & r[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3] \\ & \ \ = m_1m_2m_3 - (m_1 - r(A_1))(m_2 - r(A_2))(m_3 - r(A_3)), \end{align}$

(3.34)

$\begin{align} & r[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}] \\ & \ \ = m_1m_2m_3 - (m_1 - r(A_1))(m_2 - r(A_2))r(A_3), \end{align}$

(3.35)

$\begin{align} & r[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3] \\ & \ \ = m_1m_2m_3 - (m_1 - r(A_1))r(A_2)(m_3 - r(A_3)), \end{align}$

(3.36)

$\begin{align} & r[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3] \\ & \ \ = m_1m_2m_3 - r(A_1)(m_2 - r(A_2))(m_3 - r(A_3)), \end{align}$

(3.37)

$\begin{align} & r[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}] \\ & \ \ = m_1m_2m_3 - (m_1 - r(A_1))r(A_2)r(A_3), \end{align}$

(3.38)

$\begin{align} & r[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}] \\ & \ \ = m_1m_2m_3 - r(A_1)(m_2 - r(A_2))r(A_3), \end{align}$

(3.39)

$\begin{align} & r[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3] \\ & \ \ = m_1m_2m_3 - r(A_1)r(A_2)(m_3 - r(A_3)), \end{align}$

(3.40)

$\begin{align} & r[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}] \\ & = m_1m_2m_3 - r(A_1)r(A_2)r(A_3), \end{align}$

(3.41)

the following eight dimension equalities hold $:$

$\begin{align} & \dim( {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)) = r(A_1)r(A_2)r(A_3), \end{align}$

(3.42)

$\begin{align} & \dim( {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)) = r(A_1)r(A_2)(m_3 - r(A_3)), \end{align}$

(3.43)

$\begin{align} & \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) = r(A_1)(m_2 - r(A_2)) r(A_3), \end{align}$

(3.44)

$\begin{align} & \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) = (m_1 - r(A_1))r(A_2)r(A_3), \end{align}$

(3.45)

$\begin{align} & \dim( {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)) = r(A_1)(m_2 - r(A_2))(m_3 - r(A_3)), \end{align}$

(3.46)

$\begin{align} & \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)) = (m_1 - r(A_1))r(A_2)(m_3 - r(A_3)), \end{align}$

(3.47)

$\begin{align} & \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) = (m_1 - r(A_1))(m_2 - r(A_2))r(A_3), \end{align}$

(3.48)

$\begin{align} & \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)) = (m_1 - r(A_1))(m_2 - r(A_2))(m_3 - r(A_3)), \end{align}$

(3.49)

and the following dimension equality holds $:$

$\begin{align} & \dim( {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)) + \dim( {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)) \\ & \ \ + \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) + \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) \\ & \ \ + \dim( {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)) + \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)) \\ & \ \ + \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) + \dim( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3) \\ & = m_1m_2m_3. \end{align}$

(3.50)

${\rm (c)}$ The following eight groups of range equalities hold $:$

$\begin{align} {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3) & = {\mathscr R}(X_1X_2X_3) = {\mathscr R}(A_1 \otimes A_2\otimes A_3), \end{align}$

(3.51)

$\begin{align} {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3) & = {\mathscr R}(X_1X_2E_{X_3}) = {\mathscr R}(A_1 \otimes A_2 \otimes E_{A_3}), \end{align}$

(3.52)

$\begin{align} {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}({X_2})\cap {\mathscr R}(X_3) & = {\mathscr R}(X_1E_{X_2}X_3) = {\mathscr R}(A_1 \otimes E_{A_2}\otimes A_3), \end{align}$

(3.53)

$\begin{align} {\mathscr R}^{\perp}({X_1})\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3) & = {\mathscr R}(E_{X_1}X_2X_3) = {\mathscr R}(E_{A_1} \otimes A_2\otimes A_3), \end{align}$

(3.54)

$\begin{align} {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3) & = {\mathscr R}(X_1E_{X_2}E_{X_3}) = {\mathscr R}(A_1 \otimes E_{A_2}\otimes E_{A_3}), \end{align}$

(3.55)

$\begin{align} {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3) & = {\mathscr R}(E_{X_1}X_2E_{X_3}) = {\mathscr R}(E_{A_1} \otimes A_2 \otimes E_{A_3}), \end{align}$

(3.56)

$\begin{align} {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}({X_2})\cap {\mathscr R}(X_3) & = {\mathscr R}(X_1E_{X_2}X_3) = {\mathscr R}(E_{A_1} \otimes E_{A_2} \otimes A_3), \end{align}$

(3.57)

$\begin{align} {\mathscr R}^{\perp}({X_1})\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3) & = {\mathscr R}(E_{X_1}E_{X_2}E_{X_3}) = {\mathscr R}(E_{A_1} \otimes E_{A_2}\otimes E_{A_3}), \end{align}$

(3.58)

and the following direct sum equality holds $:$

$\begin{align} & ( {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)) \oplus ( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)) \\ & \oplus ( {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) \oplus ( {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)) \\ & \oplus ( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)) \oplus ( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)) \\ & \oplus( {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)) \oplus ( {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)) \\ & = {\mathbb C}^{m_1m_2m_3}. \end{align}$

(3.59)

${\rm (d)}$ The following eight orthogonal projector equalities hold $:$

$\begin{align} P_{ {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)} & = P_{A_1}\otimes P_{A_2} \otimes P_{A_3}, \end{align}$

(3.60)

$\begin{align} P_{ {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)} & = P_{A_1}\otimes P_{A_2} \otimes E_{A_3}, \end{align}$

(3.61)

$\begin{align} P_{ {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)} & = P_{A_1}\otimes E_{A_2} \otimes P_{A_3}, \end{align}$

(3.62)

$\begin{align} P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)} & = E_{A_1}\otimes P_{A_2} \otimes P_{A_3}, \end{align}$

(3.63)

$\begin{align} P_{ {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)} & = P_{A_1}\otimes E_{A_2} \otimes E_{A_3}, \end{align}$

(3.64)

$\begin{align} P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)} & = E_{A_1}\otimes P_{A_2} \otimes E_{A_3}, \end{align}$

(3.65)

$\begin{align} P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)} & = E_{A_1}\otimes E_{A_2} \otimes P_{A_3}, \end{align}$

(3.66)

$\begin{align} P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)} & = E_{A_1}\otimes E_{A_2} \otimes E_{A_3}, \end{align}$

(3.67)

and the following orthogonal projector equality holds $:$

$\begin{align} & P_{ {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)} + P_{ {\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)} \\ & \ \ + P_{ {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)} + P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3)} \\ & \ \ + P_{ {\mathscr R}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)} + P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}^{\perp}(X_3)} \\ & \ \ + P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}(X_3)} + P_{ {\mathscr R}^{\perp}(X_1)\cap {\mathscr R}^{\perp}(X_2)\cap {\mathscr R}^{\perp}(X_3)} = I_{m_1m_2m_3}. \end{align}$

(3.68)

${\rm (e)}$ The following eight orthogonal projector equalities hold $:$

$\begin{align} P_{[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3]} & = I_{m_1m_2m_3} - E_{A_1} \otimes E_{A_2} \otimes E_{A_3}, \end{align}$

(3.69)

$\begin{align} P_{[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}]} & = I_{m_1m_2m_3} - E_{A_1} \otimes E_{A_2} \otimes P_{A_3}, \end{align}$

(3.70)

$\begin{align} P_{[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3]} & = I_{m_1m_2m_3} - E_{A_1} \otimes P_{A_2} \otimes E_{A_3}, \end{align}$

(3.71)

$\begin{align} P_{[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3]} & = I_{m_1m_2m_3} - P_{A_1} \otimes E_{A_2} \otimes E_{A_3}, \end{align}$

(3.72)

$\begin{align} P_{[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}]} & = I_{m_1m_2m_3} - E_{A_1} \otimes P_{A_2} \otimes P_{A_3}, \end{align}$

(3.73)

$\begin{align} P_{[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}]} & = I_{m_1m_2m_3} - P_{A_1} \otimes E_{A_2} \otimes P_{A_3}, \end{align}$

(3.74)

$\begin{align} P_{[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3]} & = I_{m_1m_2m_3} - P_{A_1} \otimes P_{A_2} \otimes E_{A_3}, \end{align}$

(3.75)

$\begin{align} P_{[E_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes E_{A_2} \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes E_{A_3}]} & = I_{m_1m_2m_3} - P_{A_1} \otimes P_{A_2} \otimes P_{A_3} . \end{align}$

(3.76)

Proof. By (1.1)–(1.3),

$\begin{align*} P_{X_1} & = (A_1 \otimes I_{m_2} \otimes I_{m_3})(A_1 \otimes I_{m_2} \otimes I_{m_3})^{\dagger} \\ & = (A_1 \otimes I_{m_2} \otimes I_{m_3})(A_1^{\dagger} \otimes I_{m_2} \otimes I_{m_3}) \\ & = (A_1A_1^{\dagger}) \otimes I_{m_2} \otimes I_{m_3} \\ & = P_{A_1} \otimes I_{m_2} \otimes I_{m_3}, \end{align*}$

thus establishing the first equality in (3.29). The second and third equalities in (3.29) can be shown in a similar way. Also by (1.1)–(1.3),

$\begin{align} P_{A_1 \otimes A_2 \otimes A_3} & = (A_1 \otimes A_2 \otimes A_3)(A_1 \otimes A_2 \otimes A_3)^{\dagger} \\ & = (A_1 \otimes A_2 \otimes A_3)(A_1^{\dagger} \otimes A_2^{\dagger} \otimes A_3^{\dagger}) \\ & = (A_1A_1^{\dagger}) \otimes (A_2A_2^{\dagger}) \otimes (A_3A_3^{\dagger}) \\ & = P_{A_1} \otimes P_{A_2} \otimes P_{A_3}, \end{align}$

(3.77)

and by (1.2) and (3.29),

$\begin{align} P_{X_1}P_{X_2}P_{X_3} = (P_{A_1} \otimes I_{m_2} \otimes I_{m_3})(I_{m_1} \otimes P_{A_2} \otimes I_{m_3})(I_{m_1} \otimes I_{m_2} \otimes P_{A_3}) = P_{A_1} \otimes P_{A_2} \otimes P_{A_3}. \end{align}$

(3.78)

Combining (3.77) and (3.78) leads to (3.33).

By (2.1), (1.2)–(1.4) and (3.2),

$\begin{align*} & r[A_1 \otimes I_{m_2} \otimes I_{m_3}, \, I_{m_1} \otimes A_2 \otimes I_{m_3}, \, I_{m_1} \otimes I_{m_2} \otimes A_3] \\ & = r(A_1 \otimes I_{m_2} \otimes I_{m_3}) + r((I_{m_1} - A_1A_1^{\dagger}) \otimes[A_2 \otimes I_{m_3}, I_{m_2} \otimes A_3]) \nonumber \\ & = m_2m_3r(A_1) + r(I_{m_1} - A_1A_1^{\dagger})r[A_2 \otimes I_{m_3}, I_{m_2} \otimes A_3] \nonumber \\ & = m_2m_3r(A_1) + (m_1 - r(A_1)) (m_2m_3 - (m_2 - r(A_2))(m_3 - r(A_3))) \nonumber \\ & = m_1m_2m_3 - (m_1 - r(A_1))(m_2 - r(A_2))(m_3 - r(A_3)), \end{align*}$

thus establishing (3.34). Equations (3.35)–(3.41) can be established in a similar way.

By (3.11), we are able to obtain

${\mathscr R}(X_1)\cap {\mathscr R}(X_2) = {\mathscr R}(X_1X_2) = {\mathscr R}(X_2X_1) = {\mathscr R}(A_1 \otimes A_2\otimes I_{m_3}).$

Consequently,

${\mathscr R}(X_1)\cap {\mathscr R}(X_2)\cap {\mathscr R}(X_3) = {\mathscr R}(X_1X_2)\cap {\mathscr R}(X_3) = {\mathscr R}(X_1X_2X_3) = {\mathscr R}(A_1 \otimes A_2\otimes A_3),$

as required for (3.51). Equations (3.52)–(3.58) can be established in a similar way. Adding (3.51)–(3.58) leads to (3.59).

Taking the dimensions of both sides of (3.51)–(3.58) and applying (1.4), we obtain (3.42)–(3.50).

Equations (3.60)–(3.68) follow from (3.51)–(3.59).

Equations (3.69)–(3.77) follow from (2.6) and (3.30)–(3.32). □

In addition to (3.28), we can construct the following three dilation expressions

$\begin{align} Y_1 = I_{m_1} \otimes A_2 \otimes A_3, \ \ Y_2 = A_1 \otimes I_{m_2} \otimes A_3 \ \ and \ \ Y_3 = A_1 \otimes A_2 \otimes I_{m_3} \end{align}$

(3.79)

from any three matrices $A_1 \in {\mathbb C}^{m_1 \times n_1}, \, A_2 \in {\mathbb C}^{m_2 \times n_2}$ and $A_3 \in {\mathbb C}^{m_3 \times n_3}$ . Some concrete topics on rank equalities for the dilation expressions under vector situations were considered in ^[26]. Below, we give a sequence of results related to the three dilation expressions.

Theorem 3.4. Let $Y_1,$ $Y_2$ and $Y_3$ be the same as given in (3.79) $.$ Then $,$ we have the following results $.$

${\rm (a)}$ The following three projector equalities hold $:$

$\begin{align} P_{Y_1} = I_{m_1} \otimes P_{A_2} \otimes P_{A_3}, \ \ P_{Y_2} = P_{A_1} \otimes I_{m_2} \otimes P_{A_3} \ \ and \ \ P_{Y_3} = P_{A_1} \otimes P_{A_2} \otimes I_{m_3}. \end{align}$

(3.80)

${\rm (b)}$ The following twelve matrix equalities hold $:$

$\begin{align} P_{Y_1}P_{Y_2} & = P_{Y_2}P_{Y_1} = P_{Y_1}P_{Y_3} = P_{Y_3}P_{Y_1} = P_{Y_2}P_{Y_3} = P_{Y_3}P_{Y_2} \\ & = P_{Y_1}P_{Y_2}P_{Y_3} = P_{Y_1}P_{Y_3} P_{Y_2} = P_{Y_2}P_{Y_1} P_{Y_3} = P_{Y_2}P_{Y_3} P_{Y_1} \\ & = P_{Y_3}P_{Y_1} P_{Y_2} = P_{Y_3}P_{Y_2} P_{Y_1} = P_{A_1} \otimes P_{A_2} \otimes P_{A_3}. \end{align}$

(3.81)

${\rm (c)}$ The following rank equality holds $:$

$\begin{align} r[Y_1, \, Y_2, \, Y_3] & = m_1r(A_2)r(A_3) + m_2r(A_1)r(A_3) + m_3r(A_1)r(A_2) - 2r(A_1)r(A_2)r(A_3). \end{align}$

(3.82)

${\rm (d)}$ The following range equality holds $:$

$\begin{align} {\mathscr R}(Y_1)\cap {\mathscr R}(Y_2)\cap {\mathscr R}(Y_3) & = {\mathscr R}(A_1 \otimes A_2\otimes A_3). \end{align}$

(3.83)

${\rm (e)}$ The following dimension equality holds $:$

$\begin{align} \dim( {\mathscr R}(Y_1)\cap {\mathscr R}(Y_2)\cap {\mathscr R}(Y_3)) = r(A_1)r(A_2)r(A_3). \end{align}$

(3.84)

${\rm (f)}$ The following projector equality holds $:$

$\begin{align} P_{ {\mathscr R}(Y_1)\cap {\mathscr R}(Y_2)\cap {\mathscr R}(Y_3)} = P_{A_1}\otimes P_{A_2} \otimes P_{A_3}. \end{align}$

(3.85)

${\rm (g)}$ The following projector equality holds $:$

$\begin{align} P_{[Y_1, \, Y_2, \, Y_3]} & = I_{m_1} \otimes P_{A_2} \otimes P_{A_3} + P_{A_1}\otimes I_{m_2} \otimes P_{A_3} + P_{A_1}\otimes P_{A_2} \otimes I_{m_3} - 2(P_{A_1}\otimes P_{A_2} \otimes P_{A_3}). \end{align}$

(3.86)

Proof. Equation (3.80) follows directly from (3.79), and (3.81) follows from (3.80). Since $P_{Y_1}$ , $P_{Y_2}$ and $P_{Y_3}$ are idempotent matrices, we find from (2.8) and (3.80) that

$\begin{align*} r[Y_1, \, Y_2, \, Y_3] & = r[P_{Y_1}, \, P_{Y_2}, \, P_{Y_3}] = r(P_{Y_1}) + r(P_{Y_2}) + r(P_{Y_3}) \\ & \ \ \ - r[P_{Y_1}P_{Y_2}, \, P_{Y_1}P_{Y_3}] - r[P_{Y_2}P_{Y_1}, \, P_{Y_2}P_{Y_3}] - r[P_{Y_3}P_{Y_1}, \, P_{Y_3}P_{Y_1}] \\ & \ \ \ + r[P_{Y_1}P_{Y_2}, \, P_{Y_1}P_{Y_3}, \, P_{Y_2}P_{Y_2}] \nonumber \\ & = r(P_{Y_1}) + r(P_{Y_2}) + r(P_{Y_3}) -2 r(P_{A_1} \otimes P_{A_2} \otimes P_{A_3}) \\ & = m_1r(A_2)r(A_3) + m_2r(A_1)r(A_3) + m_3r(A_1)r(A_2) - 2r(A_1)r(A_2)r(A_3), \end{align*}$

thus establishing (3.82). Equations (3.83)–(3.86) are left as exercises for the reader. □

There are some interesting consequences to Theorems 3.3 and 3.4. For example, applying the following well-known rank inequality (cf. ^[22]):

$\begin{align*} r(A + B + C) \geq r\begin{bmatrix} A \\ B \\ C \end{bmatrix} + r[A, \, B, \, C] - r(A) - r(B) - r(C) \end{align*}$

to the sums of matrices in (3.28) and (3.80) yields the two rank inequalities

$\begin{align*} & r(A_1 \otimes I_{m_2} \otimes I_{m_3} + I_{m_1} \otimes A_2 \otimes I_{m_3} + I_{m_1} \otimes I_{m_2} \otimes A_3) \\ & \geq m_1m_2r(A_3) + m_1m_3r(A_2) + m_2m_3r(A_1) - 2m_1r(A_2)r(A_3) - 2m_2r(A_1)r(A_3) \\ & \ \ \ - 2m_3r(A_1)r(A_2) + 2r(A_1)r(A_2)r(A_3) \end{align*}$

and

$\begin{align*} & r(I_{m_1} \otimes A_2 \otimes A_3 + A_1 \otimes I_{m_2} \otimes A_3 + A_1 \otimes A_2 \otimes I_{m_3}) \\ & \geq m_1r(A_2)r(A_3) + m_2r(A_1)r(A_3) + m_3r(A_1)r(A_2) - 4r(A_1)r(A_2)r(A_3), \end{align*}$

respectively, where $A_1 \in {\mathbb C}^{m_1 \times m_1}, \, A_2 \in {\mathbb C}^{m_2 \times m_2}$ and $A_3 \in {\mathbb C}^{m_3 \times m_3}$ .

4. Conclusions

We presented a new analysis of the dilation factorizations of the Kronecker products of two or three matrices, and obtained a rich variety of exact formulas and facts related to ranks, dimensions, orthogonal projectors, and ranges of Kronecker products of matrices. Admittedly, it is easy to understand and utilize these resulting formulas and facts in dealing with Kronecker products of matrices under various concrete situations. Given the formulas and facts in the previous theorems, there is no doubt to say that this study clearly demonstrates significance and usefulness of the dilation factorizations of Kronecker products of matrices. Therefore, we believe that this study can bring deeper insights into performances of Kronecker products of matrices, and thereby can lead to certain advances of enabling methodology in the domain of Kronecker products. We also hope that the findings in this resultful study can be taken as fundamental facts and useful supplementary materials in matrix theory when identifying and approaching various theoretical and computational issues associated with Kronecker products of matrices.

Moreover, the numerous formulas and facts in this article can be extended to the situations for dilation factorizations of multiple Kronecker products of matrices, which can help us a great deal in producing more impressive and useful contributions of researches related to Kronecker products of matrices and developing other relevant mathematical techniques applicable to solving practical topics. Thus, they can be taken as a reference and a source of inspiration for deep understanding and exploration of numerous performances and properties of Kronecker products of matrices.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to express their sincere thanks to anonymous reviewers for their helpful comments and suggestions.

Conflict of interest

The authors declare that they have no conflict of interest.

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