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Research article

Assessment of indoor air quality in Tunisian childcare establishments

  • Maintaining healthy indoor air quality (IAQ) in childcare settings is essential for infants and young children, as it directly impacts their early learning, development, and overall well-being. Given their vulnerability, continuous IAQ monitoring in these environments is crucial to ensuring a safe and supportive atmosphere. This study aimed to assess IAQ factors that may affect occupant health by measuring indoor concentrations of particulate matter (PM10), selected gases such as carbon dioxide (CO2) and formaldehyde (CH2O), and thermal conditions including temperature and relative humidity. Additionally, airborne microorganism levels were analyzed, and potential environmental factors influencing microbial abundance were investigated in three childcare centers in Megrine, Tunisia, across three seasonal periods. Results revealed frequent occurrences of hygrothermal discomfort and elevated levels of CO2, CH2O, and PM10, particularly in overcrowded classrooms with poor ventilation and heating. Pathogenic bacterial species, including Staphylococcus epidermidis, Staphylococcus haemolyticus, Bacillus cereus, and Bacillus licheniformis, were repeatedly detected. Significant correlations were found between bacterial abundance and environmental factors such as PM10, CO2 levels, temperature, and humidity. These findings provide valuable insights into IAQ dynamics in childcare environments, highlighting the need for improved ventilation and air quality management strategies to safeguard children's health and well-being.

    Citation: Meher Cheberli, Marwa Jabberi, Sami Ayari, Jamel Ben Nasr, Habib Chouchane, Ameur Cherif, Hadda-Imene Ouzari, Haitham Sghaier. Assessment of indoor air quality in Tunisian childcare establishments[J]. AIMS Environmental Science, 2025, 12(2): 352-372. doi: 10.3934/environsci.2025016

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  • Maintaining healthy indoor air quality (IAQ) in childcare settings is essential for infants and young children, as it directly impacts their early learning, development, and overall well-being. Given their vulnerability, continuous IAQ monitoring in these environments is crucial to ensuring a safe and supportive atmosphere. This study aimed to assess IAQ factors that may affect occupant health by measuring indoor concentrations of particulate matter (PM10), selected gases such as carbon dioxide (CO2) and formaldehyde (CH2O), and thermal conditions including temperature and relative humidity. Additionally, airborne microorganism levels were analyzed, and potential environmental factors influencing microbial abundance were investigated in three childcare centers in Megrine, Tunisia, across three seasonal periods. Results revealed frequent occurrences of hygrothermal discomfort and elevated levels of CO2, CH2O, and PM10, particularly in overcrowded classrooms with poor ventilation and heating. Pathogenic bacterial species, including Staphylococcus epidermidis, Staphylococcus haemolyticus, Bacillus cereus, and Bacillus licheniformis, were repeatedly detected. Significant correlations were found between bacterial abundance and environmental factors such as PM10, CO2 levels, temperature, and humidity. These findings provide valuable insights into IAQ dynamics in childcare environments, highlighting the need for improved ventilation and air quality management strategies to safeguard children's health and well-being.



    The notion of metric geometric mean (MGM for short) of positive definite matrices involves many mathematical areas, e.g. matrix/operator theory, geometry, and group theory. For any positive definite matrices A and B of the same size, the MGM of A and B is defined as

    AB=A1/2(A1/2BA1/2)1/2A1/2. (1.1)

    From algebraic viewpoint, AB is a unique positive solution of the Riccati equation (see [1, Ch. 4])

    XA1X=B. (1.2)

    In fact, the explicit formula (1.1) and the equation (1.2) are two equivalent ways to describe the geometric mean; see [2]. From differential-geometry viewpoint, AB is a unique midpoint of the Riemannian geodesic interpolated from A to B, called the weighted geometric mean of A and B:

    γ(t):=AtB=A1/2(A1/2BA1/2)tA1/2,0t1. (1.3)

    This midpoint is measured through a natural Riemannian metric; see a monograph [1, Ch. 6] or Section 2. The weighted MGMs (1.3) posses rich algebraic, order, and analytic properties, namely, positive homogeneity, congruent invariance, permutation invariance, self duality, monotonicity, and continuity from above; see [3, Sect. 3].

    In operator-theoretic approach, the weighted MGMs (1.3) for positive operators on a Hilbert space are Kubo-Ando means [4] in the sense that they satisfy the monotonicity, transformer inequality, continuity from above, and normalization. More precisely, AtB is the Kubo-Ando mean associated with the operator monotone function f(x)=xt on the positive half-line, e.g. [3, Sect. 3]. The geometric mean serves as a tool for deriving matrix/operators inequalities; see the original idea in [5], see also [1, Ch. 4], [3, Sect. 3], and [6]. The cancellability of the weighted MGM together with spectral theory can be applied to solve operator mean equations; see [7]. Indeed, given two positive invertible operators A and B acting on the same Hilbert space (or positive definite matrices of the same dimension), the equation AtX=B is uniquely solvable in terms of the weighted MGM in which the weight can be any nonzero real number. See more development of geometric mean theory for matrices/operators in [8,9,10].

    A series of Lawson and Lim works investigated the theory of (weighted) MGM in various frameworks. Indeed, the geometric mean (with weight 1/2) can be naturally defined on symmetric cones [11], symmetric sets [12], two-powered twisted subgroups [12,13], and Bruhat-Tits spaces [2]. The framework of reflection quasigroups (based on point-reflection geometry and quasigroup theory) [13] allows us to define weighted MGMs via geodesics, where the weights can be any dyadic rationals. On lineated symmetric spaces [14], the weights can be arbitrary real numbers, due to the density of the dyadic rationals on the real line. Their theory can be applied to solve certain symmetric word equations in two matrix letters; see [15]. Moreover, mean equations related MGMs were investigated in [13,16].

    From the formulas (1.1) and (1.3), the matrix products are the usual products between matching-dimension matrices. We can extend the MGM theory by replacing the usual product to the semi-tensor product (STP) between square matrices of general dimension. The STP was introduced by Cheng [17], so that the STP reduces to the usual matrix product in matching-dimension case. The STP keeps various algebraic properties of the usual matrix product such as the distribution over the addition, the associativity, and compatibility with transposition, inversion and scalar multiplication. The STP is a useful tool when dealing with vectors and matrices in classical and fuzzy logic, lattices and universal algebra, and differential geometry; see [18,19]. The STPs turn out to have a variety of applications in other fields: networked evolutionary games [20], finite state machines [21], Boolean networks [19,22,23], physics [24], and engineering [25]. Many authors developed matrix equations based on STPs; e.g. Sylvester-type equations [26,27,28], and a quadratic equation AXX=B [29].

    The present paper aims to develop further theory on weighted MGMs for positive definite matrices of arbitrary dimensions, where the matrix products are given by the STPs. In this case, the weights can be arbitrary real numbers; see Section 3. It turns out that this mean satisfies various properties as in the classical case. The most interesting case is when the weights belong to the unit interval. In this case, the weighted MGM satisfies the monotonicity, and the continuity from above. Then we investigate the theory when either A or B is not assumed to be invertible. In such case, the weights are restricted to be in [0,) or (,1]. We also use a continuity argument to study the weighted MGMs, in which the weights belong to [0,1]. Moreover, we prove the cancellability of the weighted MGMs, and apply to solve certain nonlinear matrix equations concerning MGMs; see Section 4. In Section 5, we apply the theory to solve the Riccati equation and certain symmetric word equations in two matrix letters. Finally, we summarize the whole work in Section 6.

    In the next section, we setup basic notation and provide preliminaries results on the Riemannian geometry of positive definite matrices, the tensor product, and the semi-tensor product.

    Throughout, let Mm,n be the set of all m×n complex matrices, and abbreviate Mn,n to Mn. Define Cn=Mn,1, the set of n-dimensional complex vectors. Denote by AT and A the transpose and conjugate transpose of a matrix A, respectively. The n×n identity matrix is denoted by In. The general linear group of n×n invertible complex matrices is denoted by GLn. The symbols Hn, and PSn stand for the vector space of n×n Hermitian matrices, and the cone of n×n positive semidefinite matrices, respectively. For a pair (A,B)Hn×Hn, the partial ordering AB means that AB lies in the positive cone PSn. In particular, APSn if and only if A0. Let us denote the set of n×n positive definite matrices by Pn. For each AHn, the strict inequality A>0 indicates that APn.

    A matrix pair (A,B)Mm,n×Mp,q is said to satisfy factor-dimension condition if n|p or p|n. In this case, we write AkB when n=kp, and AkB when p=kn.

    Recall that Mn is a Hilbert space endowed with the Hilbert-Schmidt inner product A,BHS=trAB and the associated norm AHS=(trAA)1/2. The subset Pn, which is an open subset in Hn, is a Riemannian manifold endowed with the trace Riemannian

    ds=A1/2dAA1/2HS=[tr(A1dA)2]1/2.

    If γ:[a,b]Pn is a (piecewise) differentiable path in Pn, we define the length of γ by

    L(γ)=baγ1/2(t)γ(t)γ1/2(t)HSdt.

    For each XGLn, the congruence transformation

    ΓX:PnPn,AXAX (2.1)

    is bijective and the composition ΓXγ:[a,b]Pn is another path in Pn. For any A,BPn, the distance between A and B is given by

    δHS(A,B)=inf{ L(γ):γ is a path from A to B }.

    Lemma 2.1. (e.g. [1]). For each A,BPn, there is a unique geodesic from A to B, parametrized by

    γ(t)=A1/2(A1/2BA1/2)tA1/2,0t1. (2.2)

    This geodesic is natural in the sense that δHS(A,γ(t))=tδHS(A,B) for each t.

    For any A,BPn, denote the geodesic from A to B by [A,B].

    This subsection is a brief review on tensor products and semi-tensor products of matrices. Recall that for any matrices A=[aij]Mm,n and BMp,q, their tensor product is defined by

    AB=[aijB]Mmp,nq.

    The tensor operation (A,B)AB is bilinear and associative.

    Lemma 2.2 (e.g. [3]). Let (A,B)Mm,n×Mp,q and (P,Q)Mm×Mn. Then we have

    (1). AB=0 if and only if either A=0 or B=0;

    (2). (AB)=AB;

    (3). if (P,Q)GLm×GLn, then (PQ)1=P1Q1;

    (4). if (P,Q)PSm×PSn, then PQPSmn and (PQ)1/2=P1/2Q1/2;

    (5). if (P,Q)Pm×Pn, then PQPmn;

    (6). det(PQ)=(detP)n(detQ)m.

    To define the semi-tensor product, first consider a pair (X,Y)M1,m×Cn of row and column vectors, respectively. If XkY, then we split X into X1,X2,,XnM1,k and define the STP of X and Y as

    XY=ni=1yiXiM1,k.

    If XkY, then we split Y into Y1,Y2,,YmCk and define the STP of X and Y as

    XY=mi=1xiYiCk.

    In general, for a pair (A,B)Mm,n×Mp,q satisfying the factor-dimensional condition, we define

    AB=[AiBj]m,qi,j=1,

    where Ai is i-th row of A and Bj is the j-th column of B. More generally, for an arbitrary matrix pair (A,B)Mm,n×Mp,q, we let α=lcm(n,p) and define

    AB=(AIα/n)(BIα/p)Mαmn,αqp.

    The operation (A,B)AB turns out to be bilinear, associative, and continuous.

    Lemma 2.3 (e.g. [18]). Let (A,B)Mm,n×Mp,q and (P,Q)Mm×Mn. Then we have

    (1). (AB)=BA;

    (2). if (P,Q)GLm×GLn, then (PQ)1=Q1P1;

    (3). det(PQ)=(detP)α/m(detQ)α/n where α=lcm(m,n).

    Proposition 2.4. Let AMm,XMn and S,THm.

    (1). If A0, then XAX0.

    (2). If ST, then XSXXTX.

    (3). If A>0 and XGLn, then XAX>0.

    (4). If S>T, then XSX>XTX.

    Proof. 1) Since (XAX)=XAX, we have that XAX is Hermitian. Let α=lcm(m,n) and uCα. Set v=Xu. Using Lemma 2.2, we obtain that AIα/p0 and then, by Lemma 2.3,

    u(XAX)u=(Xu)A(Xu)=v(AIα/n)v0.

    This implies that XAX0. 2) Since ST, we have ST0. Applying the assertion 1, we get X(ST)X0, i.e., XSXXTX. The proofs of the assertions 3)-4) are similar to the assertions 1)-2), respectively.

    We extend the classical weighted MGM (1.3) for a pair of positive definite matrices of different sizes as follows.

    Definition 3.1. Let (A,B)Pm×Pn. For any tR, the t-weighted metric geometric mean (MGM) of A and B is defined by

    AtB=A1/2(A1/2BA1/2)tA1/2Mα, (3.1)

    where α=lcm(m,n).

    Note that when m=n, Eq (3.1) reduces to the classical one (1.3). In particular, A0B=AIα/m, A1B=BIα/n, A1B=AB1A, and A2B=BA1B. Fundamental properties of the weighted MGMs (3.1) are as follows.

    Theorem 3.2. Let (A,B)Pm×Pn. Let r,s,tR and α=lcm(m,n). Then

    (1). Positivity: AtB>0.

    (2). Fixed-point property: AtA=A.

    (3). Positive homogeneity: c(AtB)=(cA)t(cB) for all c>0.

    (4). Congruent invariance: C(AtB)C=(CAC)t(CBC) for all CGLα.

    (5). Self duality: (AtB)1=A1tB1.

    (6). Permutation invariance: A1/2B=B1/2A. More generally, AtB=B1tA.

    (7). Affine change of parameters: (ArB)t(AsB)=A(1t)r+tsB.

    (8). Exponential law: Ar(AsB)=ArsB.

    (9). C1(AtB)=(C1A)t(C1B) for any CPm.

    (10). Left cancellability: Let Y1,Y2Pn and tR{0}. Then the equation AtY1=AtY2 implies Y1=Y2. In other words, for each t0, the map XAtX is an injective map from Pn to Pα.

    (11). Right cancellability: Let X1,X2Pm and tR{1}. Then the equation X1tB=X2tB implies X1=X2. In other words, for each t1, the map XXtB is an injective map from Pm to Pα.

    (12). Determinantal identity: det(AtB)=(detA)(1t)αm(detB)tαn.

    Proof. The positivity of t follows from Proposition 2.4(3). Properties 2 and 3 follow directly from the formula (3.1). Let γ be the natural parametrization of the geodesic [AIα/m,BIα/n] on the space Pα as discussed in Lemma 2.1. To prove the congruent invariance, let CGLα and consider the congruence transformation ΓC defined by (2.1). Then the path γC(t):=ΓC(γ(t)) joins the points γC(0)=CAC and γC(1)=CBC. By Lemma 2.1, we obtain

    C(AtB)C=ΓC(γ(t))=γC(t)=(CAC)t(CBC).

    To prove the self duality, let β(t)=(γ(t))1. By Lemma 2.2, we have β(0)=A1Iα/m and β(1)=B1Iα/n. Thus

    (AtB)1=(γ(t))1=β(t)=A1tB1.

    To prove the 6th item, define f:RR by f(t)=1t. Let δ=γf. Then δ(0)=BIα/n and δ(1)=AIα/m. By Lemma 2.1, we have δ(t)=BtA, and thus

    AtB=γ(t)=γ(f(1t))=δ(1t)=B1tA.

    To prove the 7th item, fix r,s and let t vary. Let δ(t)=(ArB)t(AsB). We have δ(0)=ArB and δ(1)=AsB. Define f:RR,f(t)=(1t)r+ts and β=γf. We obtain

    β(t)=γ((1t)r+ts)=A(1t)r+tsB

    and β(0)=ArB, β(1)=AsB. Hence, δ(t)=β(t), i.e., (ArB)t(AsB)=A(1t)r+tsB. The exponential law is derived from the 7th item as follows: Ar(AsB)=(A0B)r(AsB)=ArsB. The 9th item follows from the congruent invariance and the self duality. To prove the left cancellability, let tR{0} and suppose that AtY1=AtY2. We have by the exponential law that

    Y1Iα/n=A1Y1=A1/t(AtY1)=A1/t(AtY2)=A1Y2=Y2Iα/n.

    It follows that (Y1Y2)Iα/n=0, and thus by Lemma 2.2 we conclude that Y1=Y2. Hence, the map XAtX is injective. The right cancellability follows from the left cancellability together with the permutation inavariance. The determinantal identity follows directly from Lemmas 2.3 and 2.2:

    det(AtB)=det(A1/2)α/mdet((A1/2BA1/2)t)(detA1/2)α/m=(detA)α/m(det(A1/2BA1/2)t)=(detA)α/m(detA)tα/m(detB)tα/n=(detA)(1t)α/m(detB)tα/n.

    This finishes the proof.

    Remark 3.3. From Theorem 3.2, a particular case of the exponential law is that (Bs)r=Bsr for any B>0 and s,tR. The congruent invariance means that the operation t is invariant under the congruence transformation ΓC for any CGLα.

    Now, we focus on weighted MGMs in which the weight lies in the interval [0,1]. Let us write AkA when the matrix sequence (Ak) converges to the matrix A. If (Ak) is a sequence in Hn, the expression AkA means that (Ak) is a decreasing sequence and AkA. Recall the following well known matrix inequality:

    Lemma 3.4 (Löwner-Heinz inequality, e.g. [3]). Let S,TPSn and w[0,1]. If ST, then SwTw.

    When the weights are in [0,1], this mean has remarkable order and analytic properties:

    Theorem 3.5. Let (A,B),(C,D)Pm×Pn and w[0,1].

    (1). Monotonicity: If AC and BD, then AwBCwD.

    (2). Continuity from above: Let (Ak,Bk)Pm×Pn for all kN. If AkA and BkB, then AkwBkAwB.

    Proof. To prove the monotonicity, suppose that AC and BD. By Proposition 2.4, we have that A1/2BA1/2A1/2DA1/2. Using Lemma 3.4 and Proposition 2.4, we obtain

    AwB=A1/2(A1/2BA1/2)wA1/2A1/2(A1/2DA1/2)wA1/2=AwD.

    This shows the monotonicity of w in the second argument. This property together with the permutation invariance in Theorem 3.2 yield

    AwB=B1wAB1wC=CwBCwD.

    To prove the continuity from above, suppose that AkA and BkB. Applying the monotonicity and the positivity, we conclude that (AkwBk) is a decreasing sequence of positive definite matrices. The continuity of the semi-tensor multiplication implies that A1/2kBkA1/2k converges to A1/2BA1/2, and thus

    A1/2k(A1/2kBkA1/2k)wA1/2kA1/2(A1/2BA1/2)wA1/2.

    Hence, AkwBkAwB.

    Now, we extend the weighted MGM to positive semidefinite matrices. Indeed, when the first matrix argument is positive definite but the second one is positive semidefinite, the weights can be any nonnegative real numbers.

    Definition 3.6. Let (A,B)Pm×PSn. For any t[0,), the t-weighted MGM of A and B is defined by

    AtB=A1/2(A1/2BA1/2)tA1/2. (3.2)

    Here, we apply a convention X0=Iα for any XMα.

    This definition is well-defined since the matrix A1/2BA1/2 is positive semidefinite according to Proposition 2.4. The permutation invariance suggests the following definition.

    Definition 3.7. Let (A,B)PSm×Pn. For any t(,1], the t-weighted MGM of A and B is defined by

    AtB=B1tA=B1/2(B1/2AB1/2)1tB1/2. (3.3)

    Here, we apply the convention X0=Iα for any XMα.

    This definition is well-defined according to Definition 3.6 (since 1t0). Note that when A>0 and B>0, Definitions 3.1, 3.6, and 3.7 are coincide. Fundamental properties of the means (3.2) and (3.3) are as follows.

    Theorem 3.8. Denote α=lcm(m,n). If either

    (i) (A,B)Pm×PSn and r,s,t0, or

    (ii) (A,B)PSm×Pn and r,s,t1,

    then

    (1). Positivity: AtB0.

    (2). Positive homogeneity: c(AtB)=(cA)t(cB) for all c>0.

    (3). Congruent invariance: C(AtB)C=(CAC)t(CBC) for all CGLα.

    (4). Affine change of parameters: (ArB)t(AsB)=A(1t)r+tsB.

    (5). Exponential law: Ar(AsB)=ArsB.

    (6). Determinantal identity: det(AtB)=(detA)(1t)αm(detB)tαn.

    Proof. The proof of each assertion is similar to that in Theorem 3.2. When APSm, we consider A+ϵImPm and take limits when ϵ0+. When BPSn, we consider B+ϵInPn and take limits when ϵ0+.

    It is natural to extend the weighted MGMs of positive definite matrices to those of positive semidefinite matrices by a limit process. Theorem 3.5 (or both Definitions 3.6 and 3.7) then suggests us that the weights must be in the interval [0,1].

    Definition 3.9. Let (A,B)PSm×PSn. For any w[0,1], the w-weighted MGM of A and B is defined by

    AwB=limε0+(A+εIm)w(B+εIn)Mα, (3.4)

    where α=lcm(m,n). Here, we apply the convention X0=Iα for any XMα.

    Lemma 3.10. Definition 3.4 is well-defined. Moreover, if (A,B)PSm×PSn, then AwBPSα.

    Proof. When ε0+, the nets A+εIm and B+εIn are decreasing nets of positive definite matrices. From the monotonicity property in Theorem 3.5, the net (A+εIm)w(B+εIn) is decreasing. Since this net is also bounded below by the zero matrix, the order-completeness of the matrix space guarantees an existence of the limit (3.4). Moreover, the matrix limit is positive semidefinite.

    Fundamental properties of weighted MGMs are listed below.

    Theorem 3.11. Let (A,B),(C,D)PSm×PSn. Let w,r,s[0,1] and α=lcm(m,n). Then

    (1). Fixed-point property: AwA=A.

    (2). Positive homogeneity: c(AwB)=(cA)w(cB) for all c0.

    (3). Congruent invariance: T(AwB)T=(TAT)w(TBT) for all TGLα.

    (4). Permutation invariance: A1/2B=B1/2A. More generally, AwB=B1wA.

    (5). Affine change of parameters: (ArB)w(AsB)=A(1w)r+wsB.

    (6). Exponential law: Ar(AsB)=ArsB.

    (7). Determinantal identity: det(AwB)=(detA)(1w)αm(detB)wαn.

    (8). Monotonicity: If AC and BD, then AwBCwD.

    (9). Continuity from above: If AkPSm and BkPSn for all kN are such that AkA and BkB, then AkwBkAwB.

    Proof. When APSm and BPSn, we can consider A+εImPm and B+εInPn, and then take limits when ε0+. The 1st-7th items now follow from Theorem 3.2 (or Theorems 3.8). The 8th-9th items follow from Theorem 3.5. For the continuity from above, if AkA and BkB, then the monotonicity implies the decreasingness of the sequence AkwBk when k. Moreover,

    limkAkwBk=limklimε0+(Ak+εIm)w(Bk+εIn)=limε0+limk(Ak+εIm)w(Bk+εIn)=limε0+(A+εIm)w(B+εIn)=AwB.

    Thus, AkwBkAwB as desire.

    In particular, the congruent invariance implies the transformer inequality:

    C(AwB)C(CAC)w(CBC),C0.

    This property together with the monotonicity, the above continuity, and the fixed-point property yield that the mean w is a Kubo-Ando mean when w[0,1]. The results in this section include those for the MGM with weight 1/2 in [30].

    In this section, we apply our theory to solve certain matrix equations concerning weighted MGMs for positive definite matrices.

    Corollary 4.1. Let APm and B,XPn with AkB. Let tR{0}. Then the mean equation

    AtX=B (4.1)

    is uniquely solvable with an explicit solution X=A1/tB. Moreover, the solution varies continuously on the given matrices A and B. In particular, the geometric mean problem A1/2X=B has a unique solution X=A2B=BA1B.

    Proof. The exponential law in Theorem 3.2 implies that

    AtX=B=A1B=At(A1/tB).

    Then the left cancellability implies that Eq (4.1) has a unique solution X=A1/tB. The continuity of the solution follows from the explicit formula (3.1) and the continuity of the semi-tensor operation.

    Remark 4.2. We can investigate the Eq (4.1) when APm and BPSn. In this case, the weight t must be positive. This equation is uniquely solvable with an explicit solution X=A1/tBPSn. For simplicity in this section, we consider only the case when all given matrices are positive definite.

    Corollary 4.3. Let APm and B,XPn with AkB. Let r,s,tR.

    (1). If s0 and t1, then the mean equation

    (AsX)tA=B (4.2)

    is uniquely solvable with an explicit solution X=A1s(1t)B.

    (2). If s+tst, then the equation

    (AsX)tX=B (4.3)

    is uniquely solvable with an explicit solution X=AλB, where λ=1/(s+tst).

    (3). If s(1t)1, then the mean equation

    (AsX)tB=X (4.4)

    is uniquely solvable with an explicit solution X=AλB, where λ=t/(sts+1).

    (4). If st, then the equation

    AsX=BtX (4.5)

    is uniquely solvable with an explicit solution X=AλB, where λ=(1t)/(st).

    (5). If srs+rt1, then the equation

    (AsX)r(BtX)=X (4.6)

    is uniquely solvable with an explicit solution X=AλB, where λ=(rtr)/(srs+rt1).

    (6). If t0, then the mean equation

    (AtX)r(BtX)=X (4.7)

    is uniquely solvable with an explicit solution X=ArB.

    Proof. For the 1st assertion, using Theorem 3.2, we have

    As(1t)X=A1t(AsX)=(AsX)tA=B.

    By Corollary 4.1, we get the desire solution.

    For the 2nd item, applying Theorem 3.2, we get

    As+tstX=X(1s)(1t)A=X1t(X1sA)=(AsX)tX=B.

    Using Corollary 4.1, we obtain that X=A1s+tstB.

    For the 3rd item, the trivial case s=t=0 yields X=A0B. Assume that s,t0. From B1t(AsX)=X, we get by Theorem 3.2 and Corollary 4.1 that AsX=B11tX=Xtt1B. Consider

    B=Xt1t(AsX)=Xt1t(X1sA)=X(t1)(1s)tA=A1λX,

    where λ=t/(sts+1). Now, we can deduce the desire solution from Corollary 4.1.

    For the 4th item, the trivial case s=0 yields that the equation BtX=AIk has a unique solution X=B1tA=At1tB. Now, consider the case s0. Using Corollary 4.1, we have

    X=A1s(BtX)=(BtX)s1sA.

    Applying (4.4), we obtain that X=B(s1)/(st)A=AλB, where λ=(1t)/(st).

    For the 5th item, the case r=0 yields that the equation AsX=X has a unique solution X=A0B. Now, consider the case r0. Using Theorem 3.2 and Corollary 4.1, we get

    BtX=(AsX)1rX=Xr1r(X1sA)=X(r1)(1s)rA=Arss+1rX.

    Applying the equation (4.5), we obtain that X=AλB, where λ=rt/(srs+rt).

    For the 6th item, setting s=t in (4.6) yields the desire result.

    Remark 4.4. Note that the cases t=0 in Eqs (4.1)–(4.5) all reduce to Eq (4.1). A particular case of (4.3) when s=t=1/2 reads that the mean equation

    (AX)X=B (4.8)

    has a unique solution X=A4/3B. In Eq (4.4), when s=t=1/2, the mean equation

    (AX)B=X

    has a unique solution X=A2/3B. If r=s=t=1/2, then Eqs (4.6) or (4.7) implies that the geometric mean X=AB is a unique solution of the equation

    (AX)(BX)=X. (4.9)

    Equtions (4.8) and (4.9) were studied in [13] in the framework of dyadic symmetric sets.

    Recall that a matrix word in two letters A,BMn is an expression of the form

    W(A,B)=Ar1Bs1Ar2Bs2ArpBspArp+1

    in which the exponents ri,siR{0} for all i=1,2,,p and rp+1R. A matrix word is said to be symmetric if it is identical to its reversal. A famous symmetric matrix-word equation is the Riccati equation

    XAX=B, (5.1)

    here A,B,X are positive definite matrices of the same dimension. Indeed, in control engineering, an optimal regulator problem for a linear dynamical system reduces to an algebraic Riccati equation (under the controllability and the observability conditions) XA1XRXXR=B, where A,B are positive definite, and R is an arbitrary square matrix. A simple case R=0 yields the Riccati equation XA1X=B.

    In our context, the Riccati equation (5.1) can be written as A12X=B or X1A1=B. The case t=2 in Corollary 4.1 reads:

    Corollary 5.1. Let APm and B,XPn with AkB. Then the Riccati equation XAX=B has a unique positive solution X=A11/2B.

    More generally, consider the following symmetric word equation in two positive definite letters A,BPn with respect to the usual products:

    B=XAXAXAXAX((p+1)-terms of X, and p-terms of A)=X(AX)p,

    here pN. We now investigate such equations with respect to semi-tensor products.

    Corollary 5.2. Let APm and B,XPn with AkB.

    (1). Let pN. Then the symmetric word equation

    X(AX)p=B (5.2)

    is uniquely solvable with an explicit solution X=A11p+1B.

    (2). Let rR{1}. Then the symmetric word equation

    X(XAX)rX=B (5.3)

    is uniquely solvable with an explicit solution

    X=(A11r+1B)1/2.

    Proof. First, since pN, we can observe the following:

    X(AX)p=X1/2(X1/2AX1/2)pX1/2.

    It follows from the results in Section 3 that

    X(AX)p=X1/2(X1/2A1X1/2)pX1/2=XpA1=A1p+1X.

    According to Corollary 4.1, the equation A1p+1X=B has a unique solution X=A11p+1B.

    To solve Eq (5.3), we observe the following:

    X(XAX)rX=X(X1A1X1)rX=X2rA1=A1r+1X2.

    According to Corollary 4.1, the equation A1r+1X2=B has a unique solution X2=A11r+1B. Hence, we get the desire formula of X.

    We extend the notion of the classical weighted MGM to that of positive definite matrices of arbitrary dimensions, so that the usual matrix products are generalized to the semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs posses not only nice properties as in the classical case, e.g. the congruent invariance and the self duality, but also affine change of parameters, exponential law, and left/right cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above (according to the famous Löwner-Heinz inequality). When the matrix A or B is not assumed to be invertible, we can define AtB where the weight t is restricted to [0,) or (,1]. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. Due to the cancellability and another properties of the weighted MGM, such equations are always uniquely solvable with solutions expressed in terms of weighted MGMs. The notion of MGMs can be applied to solve certain symmetric word equations in two positive definite letters. A particular interest of the word equations in the field of control engineering is the Riccati equation in a general form involving semi-tensor products. Our results include the classical weighted MGMs of matrices as special case.

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

    This research is supported by postdoctoral fellowship of School of Science, King Mongkut's Institute of Technology Ladkrabang. The authors thank anonymous referees for their comments and suggestions on the manuscript.

    The authors declare there is no conflicts of interest.



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