The best approximation problems between the least-squares solution manifolds of two matrix equations

1.   Introduction

Matrix equationsplay important roles in structural vibration systems^[1,2], automatic control ^[3], and other fields^[4,5]. Observe that the linear matrix equation

and the linear matrix equations

have been extensively studied, and some profound results were established. The various special solutions of Eq (1.1) have been studied in ^[6,7,8,9]. The various special solutions of Eq (1.2) have been considered in ^{[10,11,12,13,14]}. Don ^[15] studied the general symmetric solution to Eq (1.1) by applying a formula for the partitioned minimum-norm reflexive generalized inverse. Dai ^[16] derived the symmetric solution of Eq (1.1) by utilizing the singular value decomposition (SVD). Sun ^[17] provided the least-squares symmetric solution for Eq (1.1) by using the SVD. Rao and Mitra ^[18] studied the common solutionof Eq (1.2). Yuan ^[19] considered the least-squares solution and the least-squares symmetric solution with the minimum-normof Eq (1.2) by applying the SVD. Obviously, the least-squares solution of the matrix equations form a linear manifold. It is known that the manifold distance is widely used in many aspects, such as in pattern recognition^[20], image recognition ^[21,22,23], and Riemannian manifolds ^{[24,25,26,27]}. There are some results about the distance between linear manifolds. For example, Kass ^[28] obtained the expression of the distance from a point $y$ to an affine subspace (linear manifold) $L = \{x|Ax = b\}$. Dupré and Kass ^[29] and Yuan ^[30] further proposed the specific formulas for the distance between two affine subspaces. Grover ^[31] derived an expression for the distance of a matrix $A$ from any unital $C^*$-subalgebra of ${\mathbb {C}} ^{n \times n}$ by the SVD. In addition, Du and Deng ^[32] established a new characterization of gaps between two subspaces of a Hilbert space. Baksalary and Trenkler ^[33] investigated the angles and distances between two given subspaces of ${\mathbb C}^n$. Scheffer and Vahrenhold ^[34,35] proposed an algorithm to approximate the geodesic distances and approximate weighted geodesic distances on a 2-manifold in ${\mathbb R}^{ 3}$, respectively. Inspired by the works of Kass ^[28] and Yuan ^[30], in this paper we will discuss the best approximation problem between two least-squares symmetric solution manifolds of Eq (1.1) and the best approximation problem between two least-squares solution manifolds of Eq (1.2). Namely, we consider the following two problems:

Problem 1. Given $A_1, B_1, C_1,$ and $D_1 \in {\mathbb R}^{ m \times n}$, find $d(L_1, L_2) = \min_{X\in L_1, Y\in L_2}\|X-Y\|,$ and find $\hat{X}\in L_1, \hat{Y}\in L_2$ such that $\|\hat{X}-\hat{Y}\| = d(L_1, L_2)$, where

${\mathbb {SR}}^{n \times n}$ stands for the set of all $n\times n$ symmetric matrices, and $\|\cdot\|$ is the Frobenius norm.

Problem 2. Given matrices $A_2, B_2, E_2, F_2 \in {\mathbb R}^{ m \times n}$ and $C_2, D_2, G_2, H_2 \in {\mathbb R}^{ n \times p}$, find $d(L_3, L_4) = \min_{X\in L_3, Y\in L_4}\|X-Y\|,$ and find $\tilde{X}\in L_3, \tilde{Y}\in L_4$ such that $\|\tilde{X}-\tilde{Y}\| = d(L_3, L_4)$, where

The paper is organized as follows. In Section 2, we introduce some important lemmas. In Sections 3 and 4, we derive the expressions of the matrices $\hat{X}, \hat{Y}, \tilde{X}, \tilde{Y}$ and present the explicit expressions for $d(L_1, L_2)$ and $d(L_3, L_4)$ of Problems 1 and 2 by utilizing the singular value decomposition and the canonical correlation decomposition (CCD). Finally, in Section 5, we provide a simple recipe for numerical computation to solve Problem 2.

2.   Preliminaries

In order to solve Problems 1 and 2, we need the following lemmas.

Lemma 2.1. ^[19] Given $A_1, B_1, C_1,$ and $D_1 \in {\mathbb R}^{ m \times n}$, let the SVDs of the matrices $A_1$ and $C_1$ be

where $\Sigma_1 = {\rm diag}(\gamma_1, \dots, \gamma_{s_1}) > 0$ (that is, $\Sigma_1 > 0$ means that $\Sigma_1$ is a symmetric positive definite matrix), $s_1 = {\rm rank}(A_1), \; \Sigma_2 = {\rm diag}(\theta_1, \dots, \theta_{t_1}) > 0, \; t_1 = {\rm rank}(C_1)$, and $U_1 = [U_{11}, U_{12}], \; V_1 = [V_{11}, V_{12}], \; P_1 = [P_{11}, P_{12}], \; Q_1 = [Q_{11}, Q_{12}]$ are orthogonal matrices with $U_{11} \in {\mathbb R}^{ m \times s_1}, \; V_{11} \in {\mathbb R}^{n \times s_1}, \; P_{11} \in {\mathbb R}^{ m \times t_1}, \; Q_{11} \in {\mathbb R}^{n \times t_1}$. Let

Then, the solution sets $L_1$ and $L_2$ can be expressed as

where

and ${W_1*(\Sigma_1 B_{11} +B_{11}^\top \Sigma_1})$ represents the Hadamard product of $W_1$ and $\Sigma_1 B_{11} +B_{11}^\top \Sigma_1$, $W_1 = [w_{ij}^{(1)}]_{s_1\times s_1}$ with $w_{ij}^{(1)} = \frac{1}{\gamma_i^2+ \gamma_j^2} \ (i, j = 1, \dots, s_1)$, $W_2 = [w_{ij}^{(2)}]_{t_1\times t_1}$ with $w_{ij}^{(2)} = \frac{1}{\theta_i^2+ \theta_j^2}\ (i, j = 1, \dots, t_1)$, and $X_{14}\in {\mathbb {SR}} ^{(n-s_1) \times (n-s_1)}$, $Y_{14}\in {\mathbb {SR}} ^{(n-t_1) \times (n-t_1)}$ are arbitrary symmetric matrices.

Lemma 2.2. ^[19] Suppose that $A_2, B_2, E_2, F_2 \in {\mathbb R}^{ m \times n}$ and $C_2, D_2, G_2, H_2 \in {\mathbb R}^{ n \times p}$. Assume that the SVDs of the matrices $A_2, C_2, E_2,$ and $G_2$ are

where $\Sigma_3 = {\rm diag}(\lambda_1, \dots, \lambda_{s_2}) > 0, \; s_2 = {\rm rank}(A_2),$ $\Sigma_4 = {\rm diag}(\rho_1, \dots, \rho_{t_2}) > 0, \; t_2 = {\rm rank}(C_2),$ $\Sigma_5 = {\rm diag}(\epsilon_1, \dots, \epsilon_{s_3}) > 0,$ $s_3 = {\rm rank}(E_2), \; \Sigma_6 = {\rm diag}(\eta_1, \dots, \eta_{t_3}) > 0,$ $t_3 = {\rm rank}(G_2)$, and $U_2 = [U_{21}, U_{22}], \; V_2 = [V_{21}, V_{22}], \; P_2 = [P_{21}, P_{22}],$ $Q_2 = [Q_{21}, Q_{22}], U_3 = [U_{31}, U_{32}], \; V_3 = [V_{31}, V_{32}],$ $P_3 = [P_{31}, P_{32}], \; Q_3 = [Q_{31}, Q_{32}]$ are orthogonal matrices with $U_{21} \in {\mathbb R}^{ m \times s_2}, \; V_{21} \in {\mathbb R}^{n \times s_2}, \; P_{21} \in {\mathbb R}^{ n \times t_2}, \; Q_{21} \in {\mathbb R}^{p \times t_2},$ $U_{31} \in {\mathbb R}^{ m \times s_3}, \; V_{31} \in {\mathbb R}^{n \times s_3}, \; P_{31} \in {\mathbb R}^{ n \times t_3},$ and $Q_{31} \in {\mathbb R}^{p \times t_3}$. Let

Then, the solution sets $L_3$ and $L_4$ can be expressed as

where

and $W_3 = [w_{ij}^{(3)}]_{s_2\times t_2}$ with $w_{ij}^{(3)} = \frac{1}{\lambda_i^2 +\rho_j^2} \; (i = 1, \dots, s_2;j = 1, \dots, t_2)$, $W_4 = [w_{ij}^{(4)}]_{s_3\times t_3}$ with $w_{ij}^{(4)} = \frac{1} {\epsilon_i^2 +\eta_j^2} \; (i = 1, \dots, s_3;j = 1, \dots, t_3)$, and $X_{24}\in {\mathbb R} ^{(n-s_2) \times (n-t_2)}$, $Y_{24}\in {\mathbb R} ^{(n-s_3) \times (n-t_3)}$ are arbitrary matrices.

Lemma 2.3. Let $F_{26}, \; F_{56}\in {\mathbb R}^{h_1 \times k_1},$ and $C_A = {\rm diag}(\alpha_1, \dots, \alpha_{h_1}) > 0, \; S_A = {\rm diag}\; (\beta_1, \dots, \beta_{h_1}) > 0$ satisfy $\alpha_i^2+\beta_i^2 = 1 \ (i = 1, \dots, h_1)$. Then,

if and only if

Proof. Let $F_{26} = [f_{ij}], F_{56} = [g_{ij}]\in {\mathbb R}^{h_1 \times k_1}$, and $T_{23} = [t_{ij}]\in {\mathbb R}^{h_1 \times k_1}$. Then,

Now, we minimize the quantities

It is easy to obtain the minimizers

(2.8) follows from (2.9) straightforwardly. □

Lemma 2.4. Suppose that the matrices $F_{12}, \; F_{15}\in {\mathbb R}^{r_1 \times h_1}$. Let $C_A = {\rm diag}(\alpha_1, \dots, \alpha_{h_1}) > 0, \; S_A = {\rm diag}(\beta_1, \dots, \beta_{h_1}) > 0$ satisfy $\alpha_i^2+\beta_i^2 = 1\; (i = 1, \dots, h_1)$. Then,

if and only if

Proof. Let $F_{12} = [f_{ij}], \; F_{15} = [k_{ij}], \; T_{12} = [t_{ij}], \; J_{12} = [q_{ij}]\in {\mathbb R}^{r_1 \times h_1}$. Then, the minimization problem of (2.10) is equivalent to

Clearly, the function of concerning variables $t_{ij}$ and $q_{ij}$ in (2.12) is

It is easy to verify that the function $\varphi_2$ attains the smallest value at

which yields

By rewriting (2.13) in matrix forms, we immediately obtain (2.11). □

Lemma 2.5. ^[36] Assume that $C_A = {\rm diag}(\alpha_1, \dots, \alpha_{h_1}) > 0, \; S_A = {\rm diag}(\beta_1, \dots, \beta_{h_1}) > 0$ with $\alpha_i^2+\beta_i^2 = 1\; (i = 1, \dots, h_1)$, and $F_{22}, F_{25}, F_{55}\in {\mathbb R}^{h_1 \times h_1}$. Then, the problem

has unique symmetric solutions $T_{22}, J_{22} \in {\mathbb R}^{h_1 \times h_1}$ with the forms

where $W_5 = [w_{ij}^{(5)}]_{h_1\times h_1}, \; w_{ij}^{(5)} = \frac{1}{\alpha^2_i \alpha^2_j-1} \; (i, j = 1, \dots, h_1)$.

Lemma 2.6. ^[36] Let $M_{22}, \; M_{25}, \; M_{52}, \; M_{55}\in {\mathbb R}^{h_2 \times h_3}$, and let $C_3 = {\rm diag} (\kappa_1, \dots, \kappa_{h_2}) > 0, \; S_3 = {\rm diag} (\sigma_1, \dots, \sigma_{h_2}) > 0$ satisfy $\kappa_i^2+\sigma_i^2 = 1\; (i = 1, \dots, h_2), \; C_4 = {\rm diag} (\delta_1, \dots, \delta_{h_3}) > 0$, and $S_4 = {\rm diag} (\zeta_1, \dots, \zeta_{h_3}) > 0$ satisfy $\delta_i^2+\zeta_i^2 = 1\; (i = 1, \dots, h_3)$. Then,

if and only if

where $W_6 = [w_{ij}^{(6)}]_{h_2\times h_3}$ with $w_{ij}^{(6)} = \frac{1}{\kappa_i^2 \delta_j^2-1} \; (i = 1, \dots, h_2;j = 1, \dots, h_3)$.

3.   The solution of Problem 1

Let $V_{12} \in {\mathbb {R}}^{n \times (n-s_1)}$ and $Q_{12} \in {\mathbb {R}}^{n \times (n-t_1)}$ be column orthogonal matrices, and assume that $n-s_1 = {\rm rank}(V_{12}) \geq {\rm rank}(Q_{12}) = n-t_1$. Let the CCD ^[37] of the matrix pair $[V_{12}, Q_{12}]$ be

in which $F\in {\mathbb {R}}^{n \times n}, \; N_1\in {\mathbb {R}}^{(n-s_1) \times (n-s_1)},$ and $N_2\in {\mathbb {R}}^{(n-t_1) \times (n-t_1)}$ are orthogonal matrices, and

${\rm rank}(V_{12}) = r_1+h_1+k_1, \ h_1 = {\rm rank}([V_{12}, Q_{12}])+{\rm rank}(Q_{12}^\top V_{12}) -{\rm rank}(V_{12})-{\rm rank}(Q_{12}), \ r_1 = {\rm rank}(V_{12})+{\rm rank}(Q_{12})-{\rm rank}([V_{12}, Q_{12}]), \ k_1 = {\rm rank}(V_{12})- {\rm rank}(Q_{12}^\top V_{12}), \ f_1 = n-t_1-r_1-h_1$, and

with

It follows from (2.1) and (3.1) that for any $X\in L_1$ and $Y\in L_2$, we have

Write

Then,

It follows from (3.2)–(3.4) that $d(L_1, L_2) = \min_{X\in L_1, Y\in L_2}\|X-Y\|$ if and only if

From (3.5), we obtain

where $J_{11} \in {\mathbb {SR}} ^{r_1 \times r_1}$ is an arbitrary matrix. By (3.6) and (3.7), we can find that

By applying Lemma 2.3, from (3.8) we have

Solving the minimization problem (3.9) by using Lemma 2.4, we obtain

Applying Lemma 2.5, we can obtain (2.14) from (3.10). Inserting (2.14) and (3.11)–(3.14) into (3.3) and (3.4), we obtain

The relation of (3.15) can be equivalently written as

As a summary of the above discussion, we have proved the following result.

Theorem 3.1. Given the matrices $A_1, B_1, C_1,$ and $D_1 \in {\mathbb R}^{ m \times n},$ the explicit expression for $d(L_1, L_2)$ of Problem 1 can be expressed as $(3.16),$ and the matrices $\hat{X}$ and $\hat{Y}$ are given by

where

$J_{11} \in {\mathbb {SR}} ^{r_1 \times r_1}$ is an arbitrary matrix, and $X_1, Y_1, T_{22}, J_{22}$ are given by $(2.2), (2.3),$ and $(2.14)$, respectively.

4.   The solution of Problem 2

Suppose that $V_{22} \in {\mathbb{R}}^{n \times (n-s_2)}, \; V_{32} \in {\mathbb{R}}^{n \times (n-s_3)}, \; P_{22} \in {\mathbb{R}}^{n \times (n-t_2)},$ and $P_{32} \in {\mathbb{R}}^{n \times (n-t_3)}$ are column orthogonal matrices, and assume that $n-s_2 = {\rm rank}(V_{22}) \geq {\rm rank}(V_{32}) = n-s_3$ and $n-t_2 = {\rm rank}(P_{22}) \geq {\rm rank}(P_{32}) = n-t_3$. Let the CCDs of the matrix pairs $[V_{22}, V_{32}]$ and $[P_{22}, P_{32}]$ be

in which $M\in {\mathbb {R}}^{n \times n}, \; W\in {\mathbb {R}}^{n \times n}, \; N_3\in {\mathbb {R}}^{(n-s_2) \times (n-s_2)},$ $N_4\in {\mathbb {R}}^{(n-s_3) \times (n-s_3)}, \; N_5\in {\mathbb {R}}^{(n-t_2) \times (n-t_2)},$ and $N_6\in {\mathbb {R}}^{(n-t_3) \times (n-t_3)}$ are all orthogonal matrices, and

${\rm rank}(V_{22}) = r_2+h_2+k_2,$ $h_2 = {\rm rank}([V_{22}, V_{32}])+{\rm rank}(V_{32}^\top V_{22})-{\rm rank}(V_{22})-{\rm rank}(V_{32}),$ $r_2 = {\rm rank}(V_{22})+{\rm rank}(V_{32})-{\rm rank}([V_{22}, V_{32}]),$ $k_2 = {\rm rank}(V_{22})-{\rm rank}(V_{32}^\top V_{22}), \ f_2 = n-s_3-r_2-h_2$, and

with

${\rm rank}(P_{22}) = r_3+h_3+k_3, \ h_3 = {\rm rank}([P_{22}, P_{32}])+{\rm rank}(P_{32}^\top P_{22}) -{\rm rank}(P_{22})-{\rm rank}(P_{32}), \ r_3 = {\rm rank}(P_{22})+{\rm rank}(P_{32})-{\rm rank}([P_{22}, P_{32}]), \ k_3 = {\rm rank}(P_{22})-{\rm rank}(P_{32}^\top P_{22}), \ f_3 = n-t_3-r_3-h_3$, and

with

It follows from (2.5) and (4.1) that, for any $X\in L_3$ and $Y\in L_4$, we have

If we set

then

It follows from (4.2)–(4.4) that $d(L_3, L_4) = \min_{X\in L_3, Y\in L_4}\|X-Y\|$ if and only if

From (4.5), we obtain

where $Z_{11} \in {\mathbb R} ^{r_2 \times r_3}$ is an arbitrary matrix. By (4.6) and (4.7), we get

It follows from (4.8), (4.9), and Lemma 2.3 that

By using Lemma 2.4, solving minimization problems (4.10) and (4.11) yields

The relation of (2.15) follows from Lemma 2.6 and (4.12). Substituting (2.15) and (4.13)–(4.16) into (4.3) and (4.4) leads to

The relation of (4.17) can be equivalently written as

By now, we have proved the following theorem.

Theorem 4.1. Given $A_2, B_2, E_2, F_2 \in {\mathbb R}^{ m \times n}$ and $C_2, D_2, G_2, H_2 \in {\mathbb R}^{ n \times p},$ $d(L_3, L_4)$ can be expressed by $(4.18)$ and the matrices $\tilde{X}$ and $\tilde{Y}$ are given by

where

${Z_{11} \in {\mathbb R} ^{r_2 \times r_3}}$ is an arbitrary matrix, and $X_2, Y_2, R_{22}, Z_{22}$ are given by $(2.6), (2.7),$ and $(2.15)$, respectively.

5.   Numerical algorithm and numerical example

According to Theorem 4.1, we have the following algorithm to solve Problem 2.

Algorithm 1.

1) Input matrices $A_2, B_2, C_2, D_2, E_2, F_2, G_2,$ and $H_2$.

2) Compute the SVDs of the matrices $A_2, C_2, E_2,$ and $G_2$ according to (2.4).

3) Compute the CCDs of the matrix pairs $[V_{22}, V_{23}]$ and $[P_{22}, P_{23}]$ by (4.1).

4) Calculate the matrices $M_{ij}, i, j = 1, \dots, 6$ following (4.2).

5) Randomly choose the matrix $Z_{11}$.

6) Calculate the matrices $R_{ij}, \; i, j = 1, 2, 3;\; Z_{mn}, \; m, n = 2, 3;\; Z_{12}$ and $Z_{13}$ by (2.15) and (4.13)–(4.16).

7) Compute $d(L_3, L_4), \tilde{X}_{24},$ and $\tilde{Y}_{24}$ by (4.18), (4.20), and (4.21), respectively.

8) Compute matrices $\tilde{X}$ and $\tilde{Y}$ by (4.19).

Example 5.1. Let $m = 9, \; n = 8, \; p = 7,$ and the matrices $A_2, B_2, C_2, D_2, E_2, F_2, G_2$ and $H_2$ be given by

By using Algorithm 1, the matrices $\tilde{X}$ and $\tilde{Y}$ can be calculated as

and the distance between $L_3$ and $L_4$ is $d(L_3, L_4) = 13.7969$, which implies that there is no common element between linear manifolds $L_3$ and $L_4$.

6.   Conclusions

In this paper, by utilizing the singular value decompositions and the canonical correlation decompositions of matrices, we have achieved the explicit representation for the optimal approximation distance $d(L_1, L_2)$ of linear manifolds $L_1$ and $L_2$ and the matrices $\hat{X}\in L_1, \hat{Y}\in L_2$ satisfying $\|\hat{X}-\hat{Y}\| = d(L_1, L_2)$ in Problem 1 (see Theorem 3.1), and the explicit representation for the optimal approximation distance $d(L_3, L_4)$ of linear manifolds $L_3$ and $L_4$ and the matrices $\tilde{X}\in L_3, \tilde{Y}\in L_4$ satisfying $\|\tilde{X}-\tilde{Y}\| = d(L_3, L_4)$ in Problem 2 (see Theorem 4.1). Also, we have provided a simple recipe for constructing the optimal approximation solution of Problem 2, which can serve as the basis for numerical computation. The approach is demonstrated by a simple numerical example and reasonable results are produced.

Author contributions

Yinlan Chen: Conceptualization, Methodology, Project administration, Supervision, Writing-review & editing; Yawen Lan: Investigation, Software, Validation, Writing-original draft, Writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

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

Conflict of interest

The authors declare no conflicts of interest.