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

Yinlan Chen; Yawen Lan; Yinlan Chen; Yawen Lan

doi:10.3934/math.20241019

AIMS Mathematics

2024, Volume 9, Issue 8: 20939-20955. doi: 10.3934/math.20241019

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Research article

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

Yinlan Chen ^,,
Yawen Lan

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received: 16 April 2024 Revised: 01 June 2024 Accepted: 24 June 2024 Published: 28 June 2024
MSC : 15A24, 15A60

In this paper, we will deal with the following two classes of best approximation problems about the linear manifolds: Problem 1. Given matrices $ 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 $ L_1 = \left\{{X \in {\mathbb {SR}} ^{n \times n} \left|{\ \|A_1X-B_1\| = \min}\right.} \right\} $ and $ L_2 = \left\{{Y \in {\mathbb {SR}} ^{n \times n} \left|{\ \|C_1Y-D_1\| = \min}\right.} \right\} $. 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 $ L_3 = \left\{{X \in {\mathbb {R}}^{n \times n} \left|{\ \|A_2X-B_2\|^2+||XC_2-D_2\|^2 = \min}\right.} \right\} $ and $ L_4 = \left\{{Y \in {\mathbb {R}} ^{n \times n} \left|{\ \|E_2Y-F_2\|^2+||YG_2-H_2\|^2 = \min}\right.} \right\} $. We obtain explicit formulas for $ d(L_1, L_2) $ and $ d(L_3, L_4), $ and all the matrices in question by using the singular value decompositions and the canonical correlation decompositions of matrices.
- linear manifold,
- best approximation,
- singular value decomposition,
- canonical correlation decomposition
Citation: Yinlan Chen, Yawen Lan. The best approximation problems between the least-squares solution manifolds of two matrix equations[J]. AIMS Mathematics, 2024, 9(8): 20939-20955. doi: 10.3934/math.20241019

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Abstract

In this paper, we will deal with the following two classes of best approximation problems about the linear manifolds: Problem 1. Given matrices $ 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 $ L_1 = \left\{{X \in {\mathbb {SR}} ^{n \times n} \left|{\ \|A_1X-B_1\| = \min}\right.} \right\} $ and $ L_2 = \left\{{Y \in {\mathbb {SR}} ^{n \times n} \left|{\ \|C_1Y-D_1\| = \min}\right.} \right\} $. 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 $ L_3 = \left\{{X \in {\mathbb {R}}^{n \times n} \left|{\ \|A_2X-B_2\|^2+||XC_2-D_2\|^2 = \min}\right.} \right\} $ and $ L_4 = \left\{{Y \in {\mathbb {R}} ^{n \times n} \left|{\ \|E_2Y-F_2\|^2+||YG_2-H_2\|^2 = \min}\right.} \right\} $. We obtain explicit formulas for $ d(L_1, L_2) $ and $ d(L_3, L_4), $ and all the matrices in question by using the singular value decompositions and the canonical correlation decompositions of matrices.

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