Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products

: We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) ♯ and semi-tensor products ⋉ . Indeed, for each real number t and two positive definite matrices A and B of arbitrary sizes, the t - weighted SGM A ♢ t B of A and B is a unique positive solution X of the equation


Introduction
In mathematics, we are familiar with the notion of geometric mean for positive real numbers.This notion was generalized to that for positive definite matrices of the same dimension in many ways.The metric geometric mean (MGM) of two positive definite matrices A and B is defined as This mean was introduced by Pusz and Woronowicz [1] and studied in more detail by Ando [2].Algebraically, A ♯ B is a unique solution to the algebraic Riccati equation XA −1 X = B; e.g., [3].Geometrically, A ♯ B is a unique midpoint of the Riemannian geodesic interpolated from A to B, called the weighted MGM of A and B: Remarkable properties of the mean ♯ t , where t ∈ [0, 1], are monotonicity, concavity, and upper semicontinuity (according to the famous Löwner-Heinz inequality); see, e.g., [2,4] and a survey [5,Sect. 3].Moreover, MGMs play an important role in the Riemannian geometry of the positive definite matrices; see, e.g., [6,Ch. 4].
Another kind of geometric means of positive definite matrices is the spectral geometric mean (SGM), first introduced by Fiedler and Pták [7]: (1.3) Note that the scalar consistency holds, i.e., if AB = BA, then Since the SGM is based on the MGM, the SGM satisfies many nice properties as those for MGMs, for example, idempotency, homogeneity, permutation invariance, unitary invariance, self duality, and a determinantal identity.However, the SGM does not possess the monotonicity, the concavity, and the upper semi-continuity.A significant property of SGMs is that (A ♢ B) 2 is similar to AB and, they have the same spectrum; hence, the name "spectral geometric mean".The work [7] also established a similarity relation between the MGM A ♯ B and the SGM A♢B when A and B are positive definite matrices of the same size.After that, Lee and Kim [8] investigated the t-weighted SGM, where t is an arbitrary real number: Gan and Tam [9] extended certain results of [7] to the case of the t-weighted SGMs when t ∈ [0, 1].Many research topics on the SGMs have been widely studied, e.g., [10,11].Lim [12] introduced another (weighted) geometric mean of positive definite matrices varying over Hermitian unitary matrices, including the MGM as a special case.The Lim's mean has an explicit formula in terms of MGMs and SGMs.There are several ways to extend the classical studies of MGMs and SGMs.The notion of MGMs can be defined on symmetric cones [8,13] and reflection quasigroups [14] via algebraic-geometrical perspectives.In the framework of lineated symmetric spaces [14] and reflection quasigroups equipped with a compatible Hausdorff topology, we can define MGMs of arbitrary reals weights.The SGMs were also investigated on symmetric cones in [8].These geometric means can be extended to those for positive (invertible) operators on a Hilbert space; see, e.g., [15,16].The cancellability of such means has significant applications in mean equations; see, e.g., [17,18].
Another way to generalize the means (1.2) and (1.4) is to replace the traditional matrix multiplications (TMM) by the semi-tensor products (STP) ⋉.Recall that the STP is a generalization of the TMM, introduced by Cheng [19]; see more information in [20].To be more precise, consider a matrix pair (A, B) ∈ M m,n × M p,q and let α = lcm(n, p).The STP of A and B allows the two matrices to participate the TMM through the Kronecker multiplication (denoted by ⊗) with certain identity matrices: For the factor-dimension condition n = kp, we have For the matching-dimension condition n = p, the product reduces to A ⋉ B = AB.The STP occupies rich algebraic properties as those for TMM, such as bilinearity and associativity.Moreover, STPs possess special properties that TMM does not have, for example, pseudo commutativity dealing with swap matrices, and algebraic formulations of logical functions.In the last decade, STPs were beneficial to developing algebraic state space theory, so the theory can integrate ideas and methods for finite state machines to those for control theory; see a survey in [21].
Recently, the work [22] extended the MGM notion (1.1) to any pair of positive definite matrices, where the matrix sizes satisfied the factor-dimension condition: In fact, A ♯ B is a unique positive-definite solution of the semi-tensor Riccati equation After that, the MGMs of arbitrary weight t ∈ R were studied in [23].In particular, when t ∈ [0, 1], the weighted MGMs have remarkable properties, namely, the monotonicity and the upper semi-continuity.See Section 2 for more details.The present paper is a continuation of the works [22,23].Here we investigate SGMS involving STPs.We start with the matrix mean equation: where A and B are given positive definite matrices of different sizes, t ∈ R, and X is an unknown square matrix.Here, ♯ is defined by the formula (1.5).We show that this equation has a unique positive definite solution, which is defined to be the t-weighted SGM of A and B. Another characterization of weighted SGMs are obtained in terms of certain matrix equations.It turns out that this mean satisfies various properties as in the classical case.We establish a similarity relation between the MGM and the SGM of two positive definite matrices of arbitrary dimensions.Our results generalize the work [7] and relate to the work [8].Moreover, we investigate certain matrix equations involving weighted MGMs and SGMs.
The paper is organized as follows.In Section 2, we set up basic notation and give basic results on STPs, Kronecker products, and weighted MGMs of positive definite matrices.In Section 3, we characterize the weighted SGM for positive definite matrices in terms of matrix equations, then we provide fundamental properties of weighted SGMs in Section 4. In Section 5, we investigate matrix equations involving weighted SGMs and MGMs.We conclude the whole work in Section 6.

Preliminaries
Throughout, let M m,n be the set of all m × n complex matrices and abbreviate M n,n to M n .Define C n = M n,1 as the set of n-dimensional complex vectors.Denote by A T and A * the transpose and conjugate transpose of a matrix A, respectively.The n × n identity matrix is denoted by I n .The general linear group of n×n complex matrices is denoted by GL n .Let us denote the set of n×n positive definite matrices by P n .A matrix pair (A, B) ∈ M m,n × M p,q is said to satisfy a factor-dimension condition if n | p or p | n.In this case, we write A ≻ k B when n = kp, and A ≺ k B when p = kn.

Kronecker and STPs of matrices
Recall that for any matrices A = [a i j ] ∈ M m,n and B ∈ M p,q , their Kronecker product is defined by The Kronecker operation (A, B) → A ⊗ B is bilinear and associative.

Weighted MGMs of positive definite matrices
Definition 2.4.Let (A, B) ∈ P m × P n and α = lcm(m, n).For any t ∈ R, the t-weighted MGM of A and B is defined by Lemma 2.5 ( [22]).Let (A, B) ∈ P m × P n be such that A ≺ k B, then the Riccati equation Lemma 2.6 ( [23]).Let (A, B) ∈ P m × P n and X, Y ∈ P n .Let t ∈ R and α = lcm(m, n), then (i) Positive homogeneity: For any scalars a, b, c > 0, we have c(A ♯ t B) = (cA) ♯ t (cB) and, more generally, (ii) Self duality: (iv) Consistency with scalars: (vi) Cancellability: If t 0, then the equation

Characterizations of weighted SGMs in terms of matrix equations
In this section, we define and characterize weighted SGMs in terms of certain matrix equations involving MGMs and STPs.Theorem 3.1.Let (A, B) ∈ P m × P n .Let t ∈ R and α = lcm(m, n), then the mean equation has a unique solution X ∈ P α .
Proof.Note that the matrix pair (A, X) satisfies the factor-dimension condition.Let Y = (A −1 ♯ B) t and consider Using Lemma 2.5, we obtain that Y = A −1 ♯ X.Thus, A −1 ♯ X = (A −1 ♯ B) t .For the uniqueness, let Z ∈ P α be such that A −1 ♯ Z = Y.By Lemma 2.5, we get

□
We call the matrix X in Theorem 3.1 the t-weighted SGM of A and B.
Definition 3.2.Let (A, B) ∈ P m × P n and α = lcm(m, n).For any t ∈ R, the t-weighted SGM of A and B is defined by According to Lemma 2.3, we have A ♢ t B ∈ P α .In particular, A ♢ 0 B = A⊗ I α/m and A ♢ 1 B = B⊗ I α/n .When t = 1/2, we simply write A ♢ B = A ♢ 1/2 B. The formula (3.2) implies that for any t ∈ R. Note that in the case n | m, we have i.e., Eq (3.2) reduces to the same formula (1.4) as in the classical case m = n.By Theorem 3.1, we have The following theorem provides another characterization of the weighted SGMs.
Theorem 3.3.Let (A, B) ∈ P m × P n .Let t ∈ R and α = lcm(m, n), then the following are equivalent: (ii) There exists a positive definite matrix Y ∈ P α such that Moreover, the matrix Y satisfying (3.4) is uniquely determined by To show the uniqueness, let Z ∈ P α be such that We have Z ⋉ A ⋉ Z = B ⊗ I α/n .Note that the pair (A, B ⊗ I α/n ) satisfies the factor-dimension condition.Now, Lemma 2.5 implies that Conversely, suppose there exists a matrix Y ∈ P α such that Eq (3.4) holds, then (iii) Self-duality: (iv) Unitary invariance: For any U ∈ U α , we have (vii) Left and right cancellability: For any t ∈ R − {0} and Y 1 , Y 2 ∈ P n , the equation In other words, the maps X → A ♢ t X and X → X ♢ t B are injective for any t 0, 1. (viii) (A ♢ B) 2 is positively similar to A ⋉ B i.e., there is a matrix P ∈ P α such that In particular, (A ♢ B) 2 and A ⋉ B have the same eigenvalues.
Proof.Throughout this proof, let X = A ♢ t B and Y = A −1 ♯ B. From Theorem 3.3, the characteristic equation (3.4) holds.
To prove (i), set Z = B ♢ 1−t A and W = B −1 ♯ A. By Theorem 3.3, we get It follows from Lemma 2.6(ii) that The assertion (ii) follows directly from the formulas (3.2) and (2.2): To prove the self-duality (iii), set To prove (iv), let U ∈ U α and consider W = U * ⋉ Y ⋉ U. We have and, similarly, By Theorem 3.3, we arrive at (4.1).
For the assertion (v), the assumption A ⋉ B = B ⋉ A together with Lemma 2.6 (iv) yields It follows that The determinantal identity (vi) follows directly from the formula (1.4), Lemma 2.2(iii), and Lemma 2.6(v): To prove the left cancellability, let t ∈ R − {0} and suppose that A ♢ t Y 1 = A ♢ t Y 2 .We have Taking the positive square root yields and, thus, ( Using the left cancellability of MGM (Lemma 2.6(vi)), we obtain Y 1 = Y 2 .The right cancellability follows from the left cancellability together with the permutation invariance (i).
For the assertion (viii), since Note that the matrix Y 1/2 is positive definite.Thus, (A ♢ B) 2 is positively similar to A ⋉ B, so they have the same eigenvalues.
Obviously, U is a unitary matrix.We obtain This implies that (A♢ 1−t B) 1/2 U(A ♢ t B) 1/2 is positive similar to A ♯ B. □ In general, the MGM A ♯ t B and the SGM A ♢ t B are not comparable (in the Löwner partial order).We will show that A ♯ t B and A ♢ t B coincide in the case that A and B are commuting with respect to the STP.To do this, we need a lemma.
Proof.Suppose A ⋉ B = B ⋉ A. By Lemma 2.6 and Theorem 4.1, we have Next, assume that A ♯ B = A ♢ B = X.By Lemma 2.5, we have Lemma 4.5 implies that X ⋉ Y 1/2 = Y 1/2 ⋉ X.Hence, Next, we show the equivalence between (ii) and (iii).Suppose that A ♯ B = I α , then we have In particular from Theorem 4.7, when m = n, we have that A ♢ B = I n if and only if A = B −1 , if and only if, A ♯ B = I n .This result was included in [7] and related to the work [8].

□ 4 .
Fundamental properties of weighted SGMsFundamental properties of the weighted SGMs (3.2) are as follows.