We consider the Hermitian R-conjugate generalized Procrustes problem to find Hermitian R-conjugate matrix X such that is minimum, where A_k, C_k, B_l, and D_l (k = 1,2, …, p, l = 1, …, q) are given complex matrices, and p and q are positive integers. The expression of the solution to Hermitian R-conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian R-conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian R-conjugate solution to the linear system of complex matrix equations A₁X = C₁, A₂X = C₂, …, A_pX = C_p, XB₁ = D₁, …, XB_q = D_q (p and q are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.

1. Introduction

Throughout, let 𝒞^m×n denote the set of all complex m × n matrices, ℛ^m×n the set of all real m × n matrices, and the set of all matrices in ℛ^m×n with rank r. The symbols I, , A^T, A^*, A^†, and ∥A∥, respectively, stand for the identity matrix with the appropriate size, the conjugate, the transpose, the conjugate transpose, the Moore-Penrose inverse, and the Frobenius norm of A ∈ 𝒞^m×n. For A = (a_ij), B = (b_ij) ∈ 𝒞^m×n; A*B = (a_ijb_ij) ∈ 𝒞^m×n represents the Hadamard product of A and B.

A linear model of image restoration is a matrix-vector equation is

()

where y represents the observed image, x the original true image, n additive noise, and K a blurring matrix. Image restoration is to minimize blur in an observed image, namely, recover a optimal approximation of x by given y and K, and get some statistical information of n. The process of the image restoration for the model (1) can be described as

()

where ε is a small positive parameter. It is known that

is the least squares solution of (1) with minimal norm. However, for ε = 0, the solution

to (2), that is, n = 0 in (1) is not feasible. We know if ε → 0, then the solution

converges to

. In order to obtain a solution

sufficiently near to

, we usually take small ε such that 0 < ε ≪ δ, where the error norm δ : = ∥n∥ is given.

Now we consider the generalized problem of the process of the image restoration.

Problem 1. Given 𝒮⊆𝒞^n×n, m_k, h_l, p, q being positive integers, , , E ∈ 𝒞^n×n, and k = 1, …, p, l = 1, …, q. Let

()

Find

such that

()

The constraint Procrustes problem associated with several kinds of sets 𝒮, that is, p = 1 and q = 0 in (3) has been extensively studied, such as the orthogonal Procrustes problem with 𝒮 being the set of orthogonal matrices [1], the symmetric Procrustes problem [2], (M, N)-symmetric Procrustes problem [3], Hermitian, Hermitian R-symmetric and Hermitian R-skew-symmetric Procrustes problems [4], the Procrustes problems with 𝒮 constrained to the cone of symmetric positive semidefinite and symmetric elementwise matrices [5], and the generalized Procrustes analysis [6]. The optimal approximation problem (4) is initially proposed in the processes of testing or revising given data. A preliminary estimate E of the unknown matrix X in can be obtained from experimental observation values and the information of statistical distribution.

We characterize the case A_kX = C_k, XB_l = D_l, k = 1, …, p, l = 1, …, q in Problem 1 and describe it as follows.

Problem 2. Given 𝒮⊆𝒞^n×n, m₁, …, m_p, h₁, …, h_q, p and q being positive integers, , , , F ∈ 𝒞^n×n. Let

()

When

is nonempty, find

such that

()

For important results to solve Problem 2 with different sets 𝒮, we refer to [7–14].

Motivated by the work mentioned above, in this paper we mainly discuss the above two problems associated with S being the set of Hermitian R-conjugate matrices.

Recall that an n × n complex matrix K is R-conjugate if , where R ∈ ℛ^n×n is a nontrivial involution, that is, R² = I, R ≠ ±I, which was defined in [15]. A matrix K ∈ 𝒞^n×n is Hermitian R-conjugate if K^* = K and , where R^T = R⁻¹ = R ≠ ±I. The Hermitian R-conjugate matrix is very useful in scientific computation and digital signal and image processing, its special case, for example, Hermitian Toeplitz matrix, have been studied by several authors, see [14, 16–21]. We denote the set of all n × n Hermitian R-conjugate matrices by HR_c𝒞^n×n. Let R_c𝒞^n×n, Sℛ^n×n, and ASℛ^n×n denote the set of all n × n complex R-conjugate matrices, real symmetric matrices, real skew-symmetric matrices, respectively.

This paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, we derive the expression of the unique solution to the Problem 1 with 𝒮 = HR_c𝒞^n×n. In Section 4, we establish the solvability conditions for existence and an expression of the solution for Problem 2 with 𝒮 = HR_c𝒞^n×n. In Section 5, we give examples to illustrate the results obtained in this paper.

2. Preliminaries

In order to study Problems 1 and 2 with 𝒮 = HR_c𝒞^n×n, we first give some preliminary lemmas in this section.

For a nontrivial symmetric involutory matrix R ∈ ℛ^n×n, there exist positive integers r and s such that r + s = n and an n × n orthogonal matrix

such that

()

where P ∈ ℛ^n×r and Q ∈ ℛ^n×s. The columns of P(Q) form an orthogonal basis for the eigenspace of R associated with the eigenvalue 1(−1).

Throughout this paper, we always suppose the nontrivial symmetric involutory matrix R is fixed which is given by (7).

Lemma 3 (see Theorem 2.1 in [14].)A matrix K ∈ R_c𝒞^n×n if and only if there exists H₁ ∈ ℛ^n×n such that K = UH₁U^*, and K ∈ HR_c𝒞^n×n if and only if there exists H₂ ∈ Sℛ^n×n such that K = UH₂U^*, where

()

with P and Q being the same as (7).

Lemma 4. For any matrix A ∈ 𝒞^n×n, A = A₁ ⊕ A₂ ⊕ iA₃, where

()

and ⊕ denotes the direct sum.

Proof. For B ∈ 𝒞^n×n, it is obvious that B = A₁ ⊕ A₂, where A₁ are A₂ are defined in (9). Hence, we just need to prove A = B ⊕ iA₃.

For any matrix A ∈ 𝒞^n×n, it is obvious that A = B + iA₃, where B and A₃ are defined in (9).

We prove the uniqueness of A = B + iA₃. Note that

()

that is, B, A₃ ∈ R_c𝒞^n×n. If there exist D and C₃ satisfying

()

such that A = D + iC₃, then

()

Multiplying R on the left and right side and then taking the conjugate for (12), it yields

()

implying B = D and A₃ = C₃.

So A = A₁ ⊕ A₂ ⊕ iA₃ holds, where A₁, A₂, and A₃ are defined in (9).

Lemma 5. Let the symmetric involution R ∈ ℛ^n×n be given in (7) and let U be defined as (8). Then

(i)
a matrix A ∈ 𝒞^m×n satisfies if and only if AU ∈ R^m×n,
(ii)
a matrix B ∈ 𝒞^n×l satisfies if and only if U^*B ∈ R^n×l.

Proof. (i) It yields from (7) and (8) that

()

By (14)

()

that is,

()

implying

, that is, AU ∈ R^m×n.

Conversely, if AU ∈ R^m×n, according to the proof of the necessity, we can get .

The proof of (ii) can be analogously completed according to the proof of (i).

3. The Solution to Problem 1 with 𝒮 = HR_c𝒞^n×n

We, in this section, give the explicit expression of the solution to Problem 1 with 𝒮 = HR_c𝒞^n×n. In the following, we refer to the 𝒮 = HR_c𝒞^n×n in .

According to Lemma 3, if X ∈ HR_c𝒞^n×n, then

()

where U is defined as (8) and Y ∈ Sℛ^n×n. Let

()

By Lemma 5, it is easy to verify

. We obtain

()

It follows from the unitary invariance of Frobenius norm, (21), (22), and Y ∈ Sℛ^n×n that

()

Suppose

()

then

()

We first give the following lemma which can be obtained by contrast with Lemma 2.1 in [22].

Lemma 6. Given ; therefore, the singular value decomposition (SVD) of can be described as

()

where

()

Let

, and

with d_i > 0, i = 1, …, r₁;

. Then

is consistent if and only if

()

where L

is arbitrary.

Theorem 7. Given , , and positive integers m_k, h_l, p, and q, where k = 1, …, p, l = 1, …, q, the notations A_k1, A_k2, B_l1, B_l2, C_k1, C_k2, D_l1, D_l2, S, G are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of be of the form (27) with (28). Then

()

Proof. It yields from (26) that

()

By Lemma 6,

is consistent if and only if Y has the expression of (29). Taking (29) into (17), we obtain the solution set

is (30).

Theorem 8. Given E ∈ 𝒞^n×n, the equation (4) is consistent if and only if

()

Proof. Obviously, is a closed convex set. Hence, there exists the unique element such that (4) holds. By applying Theorem 7 and the unitary invariance of Frobenius norm, for , we get

()

Then

is equivalent to

()

(34) holds if and only if

()

Substituting (35) into

, we get

is (32).

Corollary 9. if and only if

()

Remark 10. when p = 1 and q = 1 in Theorem 8, we can derive a result of Theorem 4.1 in [14].

4. The Solution to Problem 2 with 𝒮 = HR_c𝒞^n×n

We refer to 𝒮 = HR_c𝒞^n×n in in the following text. In this section, we establish necessary and sufficient conditions for the existence and the expression of . When is nonempty, we present the expression of the unique solution to (6).

It follows from (21) that the system A₁X = C₁, A₂X = C₂, …, A_pX = C_p, XB₁ = D₁, …, XB_q = D_q with unknown X ∈ HR_c𝒞^n×n is consistent if and only if there exists Y ∈ Sℛ^n×n such that SY = G. We first give the following lemma.

Lemma 11 (see Theorem 1 in [7].)Given . Let the SVD of be (27) with (28). Then the matrix equation SY = G has a symmetric solution if and only if

()

and the symmetric solution can be expressed as

()

where Z

is arbitrary.

Theorem 12. Given , , , and positive integers p and q. The notations A_k1, A_k2, B_l1, B_l2, C_k1, C_k2, D_l1, D_l2, S, G are defined as (18), (19), (20), (24), and (25), respectively. Let the SVD of be of the form (27) with (28). Then the solution set is nonempty if and only if (37) holds, in which case,

()

Proof. If the solution set is nonempty, then there exists a matrix X ∈ HR_c𝒞^n×n such that A₁X = C₁, A₂X = C₂, …, A_pX = C_p, XB₁ = D₁, …, XB_q = D_q. We know that A₁X = C₁, A₂X = C₂, …, A_pX = C_p, XB₁ = D₁, …, XB_q = D_q with X ∈ HR_c𝒞^n×n is consistent if and only if there exists Y ∈ Sℛ^n×n such that SY = G. By Lemma 11, SY = G is solvable for Y ∈ Sℛ^n×n if and only if (37) holds, and the expression of the solution is (38). Insert (38) into (17), then we obtain is of the form (39).

Conversely, assume (37) holds, according to the proof of the necessity, A₁X = C₁, A₂X = C₂, …, A_pX = C_p, XB₁ = D₁, …, XB_q = D_q is solvable for X ∈ HR_c𝒞^n×n. For , by Lemma 3, .

Theorem 13. Given F ∈ 𝒞^n×n and is nonempty. Let . Then (6) is consistent for if and only if

()

In particular, if

, then

()

Proof. By Lemma 4, F = F₁ ⊕ F₂ ⊕ iF₃, where

()

When

is nonempty, for

, we get

()

Since F₁ ∈ HR_c𝒞^n×n, F₂, F₃ ∈ R_c𝒞^n×n, by Lemma 3, we obtain U^*F₁U ∈ Sℛ^n×n, U^*F₂U, U^*F₃U ∈ ℛ^n×n. Note that U^*F₂U ∈ ASℛ^n×n, then

()

Hence,

is equivalent to

()

For the orthogonal matrix V, we get

()

Then

is solvable if and only if

()

It follows from (27) that

()

By (48) and (47) we get

()

Hence, from (39) and (49), we obtain

which can be expressed as (40).

Remark 14. When p = 1 and q = 1 in Theorem 13, we can get a conclusion of Theorem 3.1 in [14].

5. Numerical Examples

We, in this section, propose an algorithm for finding the solution of Problems 1 and 2 with 𝒮 = HR_c𝒞^n×n and give illustrative numerical examples.

Let the symmetric involutory matrix

, where

. Applying the spectral decomposition of R, we obtain the orthogonal matrix

satisfying (7) and then by (8), we get

()

Algorithm 15. (1) Input A₁, A₂, …, A_p, C₁, C₂, …, C_p, B₁, …, B_q, D₁, …, D_q;

(2) compute A_k1, A_k2, B_l1, B_l2, C_k1, C_k2, D_l1, D_l2, S, G by (18), (19), (20), (24), and (25);

(3) make the SVD of S with the form of (27) and compute W₁, W₂, V₁, V₂ by (28);

(4) if (37) holds, continue, or go to step 6;

(5) input E ∈ 𝒞^n×n, compute the solution to Problem 2 with 𝒮 = HR_c𝒞^n×n by (40);

(6) input F ∈ 𝒞^n×n, compute the solution of Problem 1 with 𝒮 = HR_c𝒞^n×n by (32).

Example 1. We consider the case of p = 1 and q = 1. Suppose A, C ∈ 𝒞^5×6, B, D ∈ 𝒞^6×4, and

()

We can easily verify the solvability condition (37) does not holds. For given E ∈ 𝒞^6×6,

()

applying Algorithm 15, we get the following results:

()

The following example is about Problem 2 with 𝒮 = HR_c𝒞^6×6, p = 1 and q = 0. We list results of comparison of the solutions computed by Algorithm 15 and MATLAB procedure X = A∖C.

Example 2. Let W_a ∈ 𝒞^8×8, V_a ∈ 𝒞^6×6 be unitary matrices,

()

and

, where D = diag (3, 2, 1, t/1, t/2, t/3) and zeros(2,6) is a 2 × 6 zeros matrix. Let

and X ∈ HR_c𝒞^6×6,

()

Then compute C(t) = A(t)X for t = 10,1, 10⁻¹, 10⁻², …, 10⁻¹³. Obviously, Problem 2 with 𝒮 = HR_c𝒞^n×n is consistent for each value of t. For matrices A, C obtained above, we first use Algorithm 15 to obtain the Hermitian R-conjugate solutions approximate to X, then compute the solutions of AX = C by MATLAB procedure X = A∖C. Let

denote the solutions computed by Algorithm 15 and X_is the solutions by MATLAB procedure X = A∖C. Let

()

Analysis of Results. As a general observation from Table 1, we find that the performance of Algorithm 15 to solve Problem 2 is very good and that of the MATLAB procedure X = A∖C is quite sensitive to the conditioning of matrix A. For t = 10,1, …, 10⁻⁴, both methods behave well. In this case we should choose MATLAB procedure X = A∖C to solve Problem 2 with 𝒮 = HR_c𝒞^n×n for it is simple. However, as some singular values of A are close to zero, the solutions X_i computed by MATLAB procedure X = A∖C do not satisfy AX = C and gradually lose the property of Hermitian R-conjugate matrix, while Algorithm 15 does it well. Hence, when A has small singular values close to zero, the Algorithm 15 predominates over MATLAB procedure X = A∖C.

Table 1. Comparison of results by Algorithm 15 and the MATLAB procedure A∖C.

t	e₁	e₂	e₄	te₁	te₂	te₃	te₄
−13	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	3 · 10⁻¹⁵	3 · 10⁻²	5 · 10⁻²	4 · 10⁻²
−12	2 · 10⁻¹⁵	1 · 10⁻¹⁵	1 · 10⁻¹⁵	3 · 10⁻¹⁵	3 · 10⁻³	5 · 10⁻³	4 · 10⁻³
−11	2 · 10⁻¹⁵	2 · 10⁻¹⁵	1 · 10⁻¹⁵	3 · 10⁻¹⁵	3 · 10⁻⁴	4 · 10⁻⁴	5 · 10⁻⁴
−10	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	3 · 10⁻¹⁵	3 · 10⁻⁵	3 · 10⁻⁵	4 · 10⁻⁵
−9	3 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	3 · 10⁻⁶	5 · 10⁻⁶	4 · 10⁻⁶
−8	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	3 · 10⁻⁷	4 · 10⁻⁷	3 · 10⁻⁷
−7	3 · 10⁻¹⁵	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	4 · 10⁻⁸	5 · 10⁻⁸	5 · 10⁻⁸
−6	2 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻⁹	4 · 10⁻⁹	4 · 10⁻⁹
−5	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁰	3 · 10⁻¹⁰	3 · 10⁻¹⁰
−4	2 · 10⁻¹⁵	1 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	3 · 10⁻¹¹	3 · 10⁻¹¹	4 · 10⁻¹¹
−3	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	3 · 10⁻¹⁵	3 · 10⁻¹²	5 · 10⁻¹²	5 · 10⁻¹²
−2	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹³	3 · 10⁻¹³	3 · 10⁻¹³
−1	2 · 10⁻¹⁵	1 · 10⁻¹⁵	2 · 10⁻¹⁵	3 · 10⁻¹⁵	3 · 10⁻¹⁴	5 · 10⁻¹⁴	5 · 10⁻¹⁴
0	2 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	4 · 10⁻¹⁵	4 · 10⁻¹⁵	6 · 10⁻¹⁵	6 · 10⁻¹⁵
1	9 · 10⁻¹⁵	2 · 10⁻¹⁵	2 · 10⁻¹⁵	1 · 10⁻¹⁴	5 · 10⁻¹⁵	6 · 10⁻¹⁵	8 · 10⁻¹⁵

6. Conclusion

In this paper, we converted the Hermitian R-conjugate generalized Procrustes problem to real symmetric Procrustes problem trickly and obtained its solution set. We also investigated the Hermitian R-conjugate solution to the linear system of complex matrix equations A₁X = C₁, A₂X = C₂, …, A_pX = C_p, XB₁ = D₁, …, XB_q = D_q and established solvable conditions and the formula for the Hermitian R-conjugate solution. Moreover, we showed the optimal approximation solution to a given matrix in the above two corresponding solution set is unique, respectively. As applications, a numerical algorithm has been given and the examples have illustrated the feasibility of the algorithm. Additionally, we can further consider the least square Hermitian R-conjugate solution of the system , k = 1,2, …, n (n is positive integer) and the corresponding optimal approximation problem.

Acknowledgments

The authors are very much indebted to the anonymous referees for their constructive and valuable comments and suggestions which greatly improved the original paper. This research was supported by the Innovation Foundation of Shanghai Municipal Education Commission (13ZZ080), the National Natural Science Foundation of China (11391240185), the Shanghai University Scientific Selection and Cultivation for Outstanding Young Teachers in special fund (sjr10009), and the Natural Science Foundation of Guangxi Province (no. 2012GXNSFBA053006).

References

1 Green B. F., The orthogonal approximation of an oblique structure in factor analysis, Psychometrika. (1952) 17, 429–440, MR0053057.
10.1007/BF02288918
PubMed Google Scholar
2 Higham N. J., The symmetric Procrustes problem, BIT Numerical Mathematics. (1988) 28, no. 1, 133–143, https://doi.org/10.1007/BF01934701, MR928441.
10.1007/BF01934701
Web of Science® Google Scholar
3 Peng J., Hu X.-Y., and Zhang L., The (M, N)-symmetric Procrustes problem, Applied Mathematics and Computation. (2008) 198, no. 1, 24–34, https://doi.org/10.1016/j.amc.2007.08.094, MR2403928.
10.1016/j.amc.2007.08.094
Web of Science® Google Scholar
4 Trench W. F., Hermitian, Hermitian R-symmetric, and Hermitian R-skew symmetric Procrustes problems, Linear Algebra and Its Applications. (2004) 387, 83–98, https://doi.org/10.1016/j.laa.2004.01.018, MR2069270.
10.1016/j.laa.2004.01.018
Web of Science® Google Scholar
5 Andersson L.-E. and Elfving T., A constrained Procrustes problem, SIAM Journal on Matrix Analysis and Applications. (1997) 18, no. 1, 124–139, https://doi.org/10.1137/S0895479894277545, MR1428204.
10.1137/S0895479894277545
Web of Science® Google Scholar
6 Gower J. C., Generalized Procrustes analysis, Psychometrika. (1975) 40, 33–51, MR0405725.
10.1007/BF02291478
Web of Science® Google Scholar
7 Dai H., On the symmetric solutions of linear matrix equations, Linear Algebra and Its Applications. (1990) 131, 1–7, https://doi.org/10.1016/0024-3795(90)90370-R, MR1057060.
10.1016/0024-3795(90)90370-R
Web of Science® Google Scholar
8 Peng Z.-Y. and Hu X.-Y., The reflexive and anti-reflexive solutions of the matrix equation AX = B, Linear Algebra and Its Applications. (2003) 375, 147–155, https://doi.org/10.1016/S0024-3795(03)00607-4, MR2013461.
10.1016/S0024-3795(03)00607-4
Web of Science® Google Scholar
9 Horn R. A., Sergeichuk V. V., and Shaked-Monderer N., Solution of linear matrix equations in a ^∗congruence class, Electronic Journal of Linear Algebra. (2005) 13, 153–156, MR2148901.
10.13001/1081-3810.1157
Web of Science® Google Scholar
10 Hua G. K., Hu X., and Zhang L., A new iterative method for the matrix equation AX = B, Applied Mathematics and Computation. (2007) 15, 1434–1441.
Google Scholar
11 Porter A. D., Solvability of the matrix equation AX = B, Linear Algebra and Its Applications. (1976) 13, no. 3, 177–184, MR0414588.
10.1016/0024-3795(76)90094-X
Web of Science® Google Scholar
12 Wang Q.-W. and Yu J., On the generalized bi (skew-) symmetric solutions of a linear matrix equation and its procrust problems, Applied Mathematics and Computation. (2013) 219, no. 19, 9872–9884, https://doi.org/10.1016/j.amc.2013.03.061, MR3055704.
10.1016/j.amc.2013.03.061
Google Scholar
13 Hanna Y. S., On the solutions of tridiagonal linear systems, Applied Mathematics and Computation. (2007) 189, no. 2, 2011–2016, https://doi.org/10.1016/j.amc.2007.01.001, MR2332153.
10.1016/j.amc.2007.01.001
Web of Science® Google Scholar
14 Dong C.-Z., Wang Q.-W., and Zhang Y.-P., On the Hermitian R-conjugate solution of a system of matrix equations, Journal of Applied Mathematics. (2012) 2012, 14, https://doi.org/10.1155/2012/398085, 398085, MR3005189.
10.1155/2012/398085
Google Scholar
15 Trench W. F., Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear Algebra and Its Applications. (2004) 377, 207–218, https://doi.org/10.1016/j.laa.2003.07.013, MR2021612.
10.1016/j.laa.2003.07.013
Web of Science® Google Scholar
16 Huckle T., Serra-Capizzano S., and Tablino-Possio C., Preconditioning strategies for Hermitian indefinite Toeplitz linear systems, SIAM Journal on Scientific Computing. (2004) 25, no. 5, 1633–1654, https://doi.org/10.1137/S1064827502416332, MR2087329.
10.1137/S1064827502416332
Web of Science® Google Scholar
17 Chan R. H., Yip A. M., and Ng M. K., The best circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Numerical Analysis. (2000) 38, no. 3, 876–896, https://doi.org/10.1137/S0036142999354083, MR1781207.
10.1137/S0036142999354083
Web of Science® Google Scholar
18 Lee A., Centro-Hermitian and skew-centro-Hermitian matrices, Linear Algebra and Its Applications. (1980) 29, 205–210, https://doi.org/10.1016/0024-3795(80)90241-4, MR562760.
10.1016/0024-3795(80)90241-4
Google Scholar
19 Oppenheim A. V., Applications of Digital Signal Processing, 1978, Prentice-Hall, Englewood Cliffs, Calif, USA.
Google Scholar
20 Ng M. K., Plemmons R. J., and Pimentel F., A new approach to constrained total least squares image restoration, Linear Algebra and Its Applications. (2000) 316, no. 1–3, 237–258, https://doi.org/10.1016/S0024-3795(00)00115-4, MR1783308.
10.1016/S0024-3795(00)00115-4
Web of Science® Google Scholar
21 Kouassi R., Gouton P., and Paindavoine M., Approximation of the Karhunen-Loève tranformation and its application to colour images, Signal Processing: Image Commu. (2001) 16, 541–551.
10.1016/S0923-5965(00)00035-7
Web of Science® Google Scholar
22 Deng Y.-B., Hu X.-Y., and Zhang L., Least squares solution of BXA^T = T over symmetric, skew-symmetric, and positive semidefinite X, SIAM Journal on Matrix Analysis and Applications. (2003) 25, no. 2, 486–494, https://doi.org/10.1137/S0895479802402491, MR2047430.
10.1137/S0895479802402491
Web of Science® Google Scholar

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The Hermitian R-Conjugate Generalized Procrustes Problem

Abstract

1. Introduction

2. Preliminaries

3. The Solution to Problem 1 with 𝒮 = HR_c𝒞^n×n

4. The Solution to Problem 2 with 𝒮 = HR_c𝒞^n×n

5. Numerical Examples

6. Conclusion

Acknowledgments

References

References

Information

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The Hermitian R-Conjugate Generalized Procrustes Problem

Abstract

1. Introduction

2. Preliminaries

3. The Solution to Problem 1 with 𝒮 = HRc𝒞n×n

4. The Solution to Problem 2 with 𝒮 = HRc𝒞n×n

5. Numerical Examples

6. Conclusion

Acknowledgments

References

References

Related

Information

3. The Solution to Problem 1 with 𝒮 = HR_c𝒞^n×n

4. The Solution to Problem 2 with 𝒮 = HR_c𝒞^n×n