The Optimization on Ranks and Inertias of a Quadratic Hermitian Matrix Function and Its Applications

Abstract

We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function 𝑄−𝑋𝑃𝑋^∗ subject to a consistent system of matrix equations 𝐴𝑋=𝐶 and 𝑋𝐵=𝐷. As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities 𝐴𝑋=𝐶,𝑋𝐵=𝐷, and 𝑋𝑃𝑋^∗=(>,<,≥,≤)𝑄 in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.

1. Introduction

Throughout this paper, we denote the complex number field by ℂ. The notations ℂ^𝑚×𝑛 and ℂ_ℎ^𝑚×𝑚 stand for the sets of all 𝑚×𝑛 complex matrices and all 𝑚×𝑚 complex Hermitian matrices, respectively. The identity matrix with an appropriate size is denoted by 𝐼. For a complex matrix 𝐴, the symbols 𝐴^∗ and 𝑟(𝐴) stand for the conjugate transpose and the rank of 𝐴, respectively. The Moore-Penrose inverse of 𝐴∈ℂ^𝑚×𝑛, denoted by 𝐴^†, is defined to be the unique solution 𝑋 to the following four matrix equations (1)(2)(3)𝐴𝑋𝐴=𝐴,𝑋𝐴𝑋=𝑋,)(𝐴𝑋^∗=𝐴𝑋,(4)(𝑋𝐴)^∗=𝑋𝐴. (1) Furthermore, 𝐿_𝐴 and 𝑅_𝐴 stand for the two projectors 𝐿_𝐴=𝐼−𝐴^†𝐴 and 𝑅_𝐴=𝐼−𝐴𝐴^† induced by 𝐴, respectively. It is known that 𝐿_𝐴=𝐿^∗_𝐴 and 𝑅_𝐴=𝑅^∗_𝐴. For 𝐴∈ℂ_ℎ^𝑚×𝑚, its inertia 𝕀_𝑛(𝐴)=𝑖₊(𝐴),𝑖₋(𝐴),𝑖₀(𝐴) (2) is the triple consisting of the numbers of the positive, negative, and zero eigenvalues of 𝐴, counted with multiplicities, respectively. It is easy to see that 𝑖₊(𝐴)+𝑖₋(𝐴)=𝑟(𝐴). For two Hermitian matrices 𝐴 and 𝐵 of the same sizes, we say 𝐴>𝐵(𝐴≥𝐵) in the Löwner partial ordering if 𝐴−𝐵 is positive (nonnegative) definite.

The investigation on maximal and minimal ranks and inertias of linear and quadratic matrix function is active in recent years (see, e.g., [1–24]). Tian [21] considered the maximal and minimal ranks and inertias of the Hermitian quadratic matrix function ℎ(𝑋)=𝐴𝑋𝐵𝑋^∗𝐴^∗+𝐴𝑋𝐶+𝐶^∗𝑋^∗𝐴^∗+𝐷, (3) where 𝐵 and 𝐷 are Hermitian matrices. Moreover, Tian [22] investigated the maximal and minimal ranks and inertias of the quadratic Hermitian matrix function 𝑓(𝑋)=𝑄−𝑋𝑃𝑋^∗ (4) such that 𝐴𝑋=𝐶.

The goal of this paper is to give the maximal and minimal ranks and inertias of the matrix function (4) subject to the consistent system of matrix equations 𝐴𝑋=𝐶,𝑋𝐵=𝐷, (5) where 𝑄∈ℂ_ℎ^𝑛×𝑛,  𝑃∈ℂ_ℎ^𝑝×𝑝 are given complex matrices. As applications, we consider the necessary and sufficient conditions for the solvability to the systems of matrix equations and inequality 𝐴𝑋=𝐶,𝑋𝐵=𝐷,𝑋𝑃𝑋^∗=𝑄,𝐴𝑋=𝐶,𝑋𝐵=𝐷,𝑋𝑃𝑋^∗>𝑄,𝐴𝑋=𝐶,𝑋𝐵=𝐷,𝑋𝑃𝑋^∗<𝑄,𝐴𝑋=𝐶,𝑋𝐵=𝐷,𝑋𝑃𝑋^∗≥𝑄,𝐴𝑋=𝐶,𝑋𝐵=𝐷,𝑋𝑃𝑋^∗≤𝑄, (6) in the Löwner partial ordering to be feasible, respectively.

2. The Optimization on Ranks and Inertias of (4) Subject to (5)

In this section, we consider the maximal and minimal ranks and inertias of the quadratic Hermitian matrix function (4) subject to (5). We begin with the following lemmas.

Lemma 1 (see [3]). Let 𝐴∈ℂ_ℎ^𝑚×𝑚,  𝐵∈ℂ^𝑚×𝑝, and 𝐶∈ℂ^𝑞×𝑚 be given and denote 𝑃₁=𝐵𝐴𝐵_∗0,𝑃₂=𝐴𝐶^∗,𝑃𝐶0₃=𝐴𝐵𝐶^∗𝐵^∗00,𝑃₄=𝐴𝐵𝐶^∗.𝐶00 (7) Then max_{𝑌∈ℂ𝑝×𝑞}𝑟𝐴−𝐵𝑌𝐶−(𝐵𝑌𝐶)^∗=min𝑟𝐴𝐵𝐶^∗,𝑟𝑃₁,𝑟𝑃₂,min_{𝑌∈ℂ𝑝×𝑞}𝑟𝐴−𝐵𝑌𝐶−(𝐵𝑌𝐶)^∗=2𝑟𝐴𝐵𝐶^∗+max𝑤₊+𝑤₋,𝑔₊+𝑔₋,𝑤₊+𝑔₋,𝑤₋+𝑔₊,max_{𝑌∈ℂ𝑝×𝑞}𝑖_±𝐴−𝐵𝑌𝐶−(𝐵𝑌𝐶)^∗=min𝑖_±𝑃₁,𝑖_±𝑃₂,min_{𝑌∈ℂ𝑝×𝑞}𝑖_±𝐴−𝐵𝑌𝐶−(𝐵𝑌𝐶)^∗=𝑟𝐴𝐵𝐶^∗+max𝑖_±𝑃₁−𝑟𝑃₃,𝑖_±𝑃₂−𝑟𝑃₄, (8) where 𝑤_±=𝑖_±𝑃₁−𝑟𝑃₃,𝑔_±=𝑖_±𝑃₂−𝑟𝑃₄. (9)

Lemma 2 (see [4]). Let 𝐴∈ℂ^𝑚×𝑛,  𝐵∈ℂ^𝑚×𝑘,  𝐶∈ℂ^𝑙×𝑛,  𝐷∈ℂ^𝑚×𝑝,  𝐸∈ℂ^𝑞×𝑛,  𝑄∈ℂ^𝑚1×𝑘, and 𝑃∈ℂ^𝑙×𝑛1 be given. Then (1)𝑟(𝐴)+𝑟𝑅_𝐴𝐵=𝑟(𝐵)+𝑟𝑅_𝐵𝐴,(2)𝑟(𝐴)=𝑟𝐴𝐵+𝑟𝐶𝐿_𝐴=𝑟(𝐶)+𝑟𝐴𝐿_𝐶𝐴𝐶,(3)𝑟(𝐵)(𝐶)𝑅=𝑟+𝑟+𝑟_𝐵𝐴𝐿_𝐶,(4)𝑟(𝑃)(𝑄)=𝑟𝐴𝐵𝐶0+𝑟+𝑟𝐴𝐵𝐿_𝑄𝑅_𝑃,(5)𝑟𝑅𝐶0=𝑟𝐴𝐵0𝐶0𝑃0𝑄0_𝐵𝐴𝐿_𝐶𝑅_𝐵𝐷𝐸𝐿_𝐶0+𝑟(𝐵)+𝑟(𝐶)=𝑟.𝐴𝐷𝐵𝐸00𝐶00 (10)

Lemma 3 (see [23]). Let 𝐴∈ℂ_ℎ^𝑚×𝑚,  𝐵∈ℂ^𝑚×𝑛,  𝐶∈ℂ_ℎ^𝑛×𝑛,  𝑄∈ℂ^𝑚×𝑛, and 𝑃∈ℂ^𝑝×𝑛 be given, and, 𝑇∈ℂ^𝑚×𝑚 be nonsingular. Then (1)𝑖_±𝑇𝐴𝑇^∗=𝑖_±(𝐴),(2)𝑖_±𝐴00𝐶=𝑖_±(𝐴)+𝑖_±(𝐶),(3)𝑖_±𝑄0𝑄_∗0(𝑄),(4)𝑖=𝑟_±𝐴𝐵𝐿_𝑃𝐿_𝑃𝐵^∗0+𝑟(𝑃)=𝑖_±𝐵𝐴𝐵0_∗0𝑃^∗.0𝑃0 (11)

Lemma 4. Let 𝐴,  𝐶,  𝐵, and 𝐷 be given. Then the following statements are equivalent.(1)System (5) is consistent.(2)Let 𝑟(𝐴),𝐷𝐵𝐴𝐶=𝑟=𝑟(𝐵),𝐴𝐷=𝐶𝐵. (12)In this case, the general solution can be written as 𝑋=𝐴^†𝐶+𝐿_𝐴𝐷𝐵^†+𝐿_𝐴𝑉𝑅_𝐵, (13) where 𝑉 is an arbitrary matrix over ℂ with appropriate size.

Now we give the fundamental theorem of this paper.

Theorem 5. Let 𝑓(𝑋) be as given in (4) and assume that 𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 in (5) is consistent. Then max_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑟𝑄−𝑋𝑃𝑋^∗=min𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗(𝐴)(𝐵)(𝑃)−𝑟−𝑟−𝑟,2𝑛+𝑟𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗(𝐴)𝐷−2𝑟,𝑟𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−2𝑟(𝐵),min_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑟𝑄−𝑋𝑃𝑋^∗=2𝑛+2𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗(𝐴)(𝐵)(𝑃)𝑠−2𝑟−2𝑟−𝑟+max₊+𝑠₋,𝑡₊+𝑡₋,𝑠₊+𝑡₋,𝑠₋+𝑡₊,max_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑖_±𝑄−𝑋𝑃𝑋^∗=min𝑛+𝑖_±𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−𝑟(𝐴),𝑖_±𝐷𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−𝑟(𝐵),min_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑖_±𝑄−𝑋𝑃𝑋^∗=𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖_±(𝑃)+max𝑠_±,𝑡_±, (14)(15) where 𝑠_±=−𝑛+𝑟(𝐴)−𝑖_∓(𝑃)+𝑖_±𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−𝑟𝐶𝑃𝐴𝑄𝐴^∗𝐵^∗𝐷^∗𝐴^∗,𝑡_±=−𝑛+𝑟(𝐴)−𝑖_∓(𝑃)+𝑖_±𝐷𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−𝐷0𝑃𝐵𝐴𝑄𝐶𝑃0_∗𝐵^∗0. (16)

Proof. It follows from Lemma 4 that the general solution of (4) can be expressed as 𝑋=𝑋₀+𝐿_𝐴𝑉𝑅_𝐵, (17) where 𝑉 is an arbitrary matrix over ℂ and 𝑋₀ is a special solution of (5). Then 𝑄−𝑋𝑃𝑋^∗=𝑄−𝑋₀+𝐿_𝐴𝑉𝑅_𝐵𝑃𝑋₀+𝐿_𝐴𝑉𝑅_𝐵^∗. (18) Note that 𝑟𝑄−𝑋₀+𝐿_𝐴𝑉𝑅_𝐵𝑃𝑋₀+𝐿_𝐴𝑉𝑅_𝐵^∗𝑄𝑋=𝑟⁰+𝐿_𝐴𝑉𝑅_𝐵𝑃𝑃𝑋₀+𝐿_𝐴𝑉𝑅_𝐵^∗𝑃−𝑟(𝑃)=𝑟𝑄𝑋₀𝑃𝑃𝑋^∗₀𝑃+𝐿_𝐴0𝑉0𝑅_𝐵𝑃+𝐿_𝐴0𝑉0𝑅_𝐵𝑃^∗(𝑃),𝑖−𝑟_±𝑄−𝑋₀+𝐿_𝐴𝑉𝑅_𝐵𝑃𝑋₀+𝐿_𝐴𝑉𝑅_𝐵^∗=𝑖_±𝑄𝑋₀+𝐿_𝐴𝑉𝑅_𝐵𝑃𝑃𝑋₀+𝐿_𝐴𝑉𝑅_𝐵^∗𝑃−𝑖_±(𝑃)=𝑖_±𝑄𝑋₀𝑃𝑃𝑋^∗₀𝑃+𝐿_𝐴0𝑉0𝑅_𝐵𝑃+𝐿_𝐴0𝑉0𝑅_𝐵𝑃^∗−𝑖_±(𝑃). (19)(20) Let 𝑞(𝑉)=𝑄𝑋₀𝑃𝑃𝑋^∗₀𝑃+𝐿_𝐴0𝑉0𝑅_𝐵𝑃+𝐿_𝐴0𝑉0𝑅_𝐵𝑃^∗. (21) Applying Lemma 1 to (19) and (20) yields max_𝑉𝑟𝑞(𝑉)=min𝑟(𝑀),𝑟𝑀₁,𝑟𝑀₂,min_𝑉𝑟𝑞(𝑉(𝑀)𝑠)=2𝑟+max₊+𝑠₋,𝑡₊+𝑡₋,𝑠₊+𝑡₋,𝑠₋+𝑡₊,max_𝑉𝑖_±𝑞(𝑉)=min𝑖_±𝑀₁,𝑖_±𝑀₂,min_𝑉𝑖_±𝑞(𝑉)=𝑟(𝑀)+max𝑠_±,𝑡_±, (22) where 𝑀=𝑄𝑋₀𝑃𝐿_𝐴0𝑃𝑋^∗₀𝑃0𝑃𝑅_𝐵,𝑀₁=𝑄𝑋₀𝑃𝐿_𝐴𝑃𝑋^∗₀𝐿𝑃0_𝐴,𝑀00₂=𝑄𝑋₀𝑃0𝑃𝑋^∗₀𝑃𝑃𝑅_𝐵0𝑅_𝐵𝑃0,𝑀₃=𝑄𝑋₀𝑃𝐿_𝐴0𝑃𝑋^∗₀𝑃0𝑃𝑅_𝐵𝐿_𝐴,𝑀000₄=𝑄𝑋₀𝑃𝐿_𝐴0𝑃𝑋^∗₀𝑃0𝑃𝑅_𝐵0𝑅_𝐵,𝑠𝑃00_±=𝑖_±𝑀₁−𝑟𝑀₃,𝑡_±=𝑖_±𝑀₂−𝑟𝑀₄. (23) Applying Lemmas 2 and 3, elementary matrix operations and congruence matrix operations, we obtain 𝑟(𝑀)=𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗(𝐴)(𝐵),𝑟𝑀−𝑟−𝑟₁=2𝑛+𝑟𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗(𝐴)(𝑃),𝑖−2𝑟+𝑟_±𝑀₁=𝑛+𝑖_±𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−𝑟(𝐴)+𝑖_±(𝑃),𝑟𝑀₂𝐷=𝑟𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0(𝐵)(𝑃),𝑖−2𝑟+𝑟_±𝑀₂=𝑖_±𝐷𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−𝑟(𝐵)+𝑖_±(𝑃),𝑟𝑀₃=2𝑛+𝑟(𝑃)−2𝑟(𝐴)−𝑟(𝐵)+𝑟𝐶𝑃𝐴𝑄𝐴^∗𝐵^∗𝐷^∗𝐴^∗,𝑟𝑀₄(𝑃)𝐷=𝑛+𝑟+𝑟0𝑃𝐵𝐴𝑄𝐶𝑃0_∗𝐵^∗0−2𝑟(𝐵)−𝑟(𝐴). (24) Substituting (24) into (22), we obtain the results.

Using immediately Theorem 5, we can easily get the following.

Theorem 6. Let 𝑓(𝑋) be as given in (4), 𝑠_± and let 𝑡_± be as given in Theorem 5 and assume that 𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 in (5) are consistent. Then we have the following.(a)𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 have a common solution such that 𝑄−𝑋𝑃𝑋^∗≥0 if and only if 𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₋(𝑃)+𝑠₋≤0,𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₋(𝑃)+𝑡₋≤0. (25)(b)𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 have a common solution such that 𝑄−𝑋𝑃𝑋^∗≤0 if and only if 𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₊(𝑃)+𝑠₊≤0,𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₊(𝑃)+𝑡₊≤0. (26)(c)𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 have a common solution such that 𝑄−𝑋𝑃𝑋^∗>0 if and only if 𝑖₊𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗(𝐴)𝑖−𝑟≥0,₊𝐷𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−𝑟(𝐵)≥𝑛. (27)(d)𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 have a common solution such that 𝑄−𝑋𝑃𝑋^∗<0 if and only if 𝑖₋𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗(𝐴)𝑖−𝑟≥0,₋𝐷𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−𝑟(𝐵)≥𝑛. (28)(e)All common solutions of 𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 satisfy 𝑄−𝑋𝑃𝑋^∗≥0 if and only if 𝑛+𝑖₋𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−𝑟(𝐴)=0,𝑜𝑟,𝑖₋𝐷𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−𝑟(𝐵)=0. (29)(f)All common solutions of 𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 satisfy 𝑄−𝑋𝑃𝑋^∗≤0 if and only if 𝑛+𝑖₊𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−𝑟(𝐴)=0,𝑜𝑟,𝑖₊𝐷𝑄0𝐷0−𝑃𝐵_∗𝐵^∗0−𝑟(𝐵)=0. (30)(g)All common solutions of 𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 satisfy 𝑄−𝑋𝑃𝑋^∗>0 if and only if 𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₊(𝑃)+𝑠₊=𝑛, (31) or 𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₊(𝑃)+𝑡₊=𝑛. (32)(h)All common solutions of 𝐴𝑋=𝐶 and  𝑋𝐵=𝐷 satisfy 𝑄−𝑋𝑃𝑋^∗<0 if and only if 𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₋(𝑃)+𝑠₋=𝑛, (33) or 𝑛+𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−𝑟(𝐴)−𝑟(𝐵)−𝑖₋(𝑃)+𝑡₋=𝑛. (34)(i)𝐴𝑋=𝐶,  𝑋𝐵=𝐷, and 𝑄=𝑋𝑃𝑋^∗ have a common solution if and only if 2𝑛+2𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−2𝑟(𝐴)−2𝑟(𝐵)−𝑟(𝑃)+𝑠₊+𝑠₋≤0,2𝑛+2𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−2𝑟(𝐴)−2𝑟(𝐵)−𝑟(𝑃)+𝑡₊+𝑡₋≤0,2𝑛+2𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−2𝑟(𝐴)−2𝑟(𝐵)−𝑟(𝑃)+𝑠₊+𝑡₋≤0,2𝑛+2𝑟0𝑃𝑃𝐴𝑄𝐶𝑃0−𝐷^∗0𝐵^∗−2𝑟(𝐴)−2𝑟(𝐵)−𝑟(𝑃)+𝑠₋+𝑡₊≤0. (35)

Let 𝑃=𝐼 in Theorem 5, we get the following corollary.

Corollary 7. Let 𝑄∈ℂ^𝑛×𝑛,  𝐴,  𝐵,  𝐶, and 𝐷 be given. Assume that (5) is consistent. Denote 𝑇₁=𝐵𝐶𝐴𝑄_∗𝐷^∗,𝑇₂=𝐴𝑄𝐴^∗−𝐶𝐶^∗,𝑇₃=𝐷𝑄𝐷_∗𝐵^∗𝐵,𝑇₄=𝐶𝐴𝑄𝐴^∗𝐵^∗𝐷^∗𝐴^∗,𝑇₅=𝐵𝐶𝐵𝐴𝑄_∗𝐵𝐷^∗. (36) Then, max_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑟𝑄−𝑋𝑋^∗=min𝑛+𝑟𝑇₁−𝑟(𝐴)−𝑟(𝐵),2𝑛+𝑟𝑇₂−2𝑟(𝐴),𝑛+𝑟𝑇₃−2𝑟(𝐵),min_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑟𝑄−𝑋𝑋^∗=2𝑟𝑇₁+max𝑟𝑇₂−2𝑟𝑇₄,−𝑛+𝑟𝑇₃−2𝑟𝑇₅,𝑖₊𝑇₂+𝑖₋𝑇₃−𝑟𝑇₄−𝑟𝑇₅,−𝑛+𝑖₋𝑇₂+𝑖₊𝑇₃−𝑟𝑇₄−𝑟𝑇₅max_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑖₊𝑄−𝑋𝑋^∗=min𝑛+𝑖₊𝑇₂−𝑟(𝐴),𝑖₊𝑇₃−𝑟(𝐵),max_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑖₋𝑄−𝑋𝑋^∗=min𝑛+𝑖₋𝑇₂−𝑟(𝐴),𝑛+𝑖₋𝑇₃−𝑟(𝐵),min_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑖₊𝑄−𝑋𝑋^∗=𝑟𝑇₁+max𝑖₊𝑇₂−𝑟𝑇₄,𝑖₊𝑇₃−𝑛−𝑟𝑇₅,min_{𝐴𝑋=𝐶,𝑋𝐵=𝐷}𝑖₋𝑄−𝑋𝑋^∗=𝑟𝑇₁+max𝑖₋𝑇₂−𝑟𝑇₄,𝑖₋𝑇₃−𝑟𝑇₅. (37)

Remark 8. Corollary 7 is one of the results in [24].

Let 𝐵 and 𝐷 vanish in Theorem 5, then we can obtain the maximal and minimal ranks and inertias of (4) subject to 𝐴𝑋=𝐶.

Corollary 9. Let 𝑓(𝑋) be as given in (4) and assume that 𝐴𝑋=𝐶 is consistent. Then max_{𝐴𝑋=𝐶}𝑟𝑄−𝑋𝑃𝑋^∗=min𝑛+𝑟𝐴𝑄𝐶𝑃−𝑟(𝐴)−𝑟(𝐵),2𝑛+𝑟𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−2𝑟(𝐴),𝑟(𝑄)+𝑟(𝑃)min_{𝐴𝑋=𝐶}𝑟𝑄−𝑋𝑃𝑋^∗(𝐴)𝑠=2𝑛+2𝑟𝐴𝑄𝐶𝑃−2𝑟+max₊+𝑠₋,𝑡₊+𝑡₋,𝑠₊+𝑡₋,𝑠₋+𝑡₊,max_{𝐴𝑋=𝐶}𝑖_±𝑄−𝑋𝑃𝑋^∗=min𝑛+𝑖_±𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−𝑟(𝐴),𝑖_±(𝑄)+𝑖_∓(𝑃),min_{𝐴𝑋=𝐶}𝑖_±𝑄−𝑋𝑃𝑋^∗=𝑛+𝑟𝐴𝑄𝐶𝑃−𝑟(𝐴)+𝑖_∓(𝑃)+max𝑠_±,𝑡_±, (38) where 𝑠_±=−𝑛+𝑟(𝐴)−𝑖_∓(𝑃)+𝑖_±𝐴𝑄𝐴^∗−𝐶𝑃𝐶^∗−𝑟𝐶𝑃𝐴𝑄𝐴^∗,𝑡_±=−𝑛+𝑟(𝐴)+𝑖_±(𝑄)−𝑟(𝑃)−𝐴𝑄𝐶𝑃. (39)

Remark 10. Corollary 9 is one of the results in [22].