Positive Definite Solutions of the Matrix Equation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.2055pt" id="M1" height="21.3863pt" version="1.1" viewBox="-0.0657574 -16.1808 165.131 21.3863" width="165.131pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"/></g><g transform="matrix(.012,0,0,-0.012,13.127,-7.578)"><path id="g50-115" d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"/></g><g transform="matrix(.017,0,0,-0.017,22.769,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,36.678,.009)"><path id="g119-65" d="M697 -46L668 -39C637 -126 611 -142 545 -142H163L474 298L206 694H483C558 694 584 688 600 668C615 651 627 625 639 580L667 585L655 739H70V714L378 254L53 -203V-228H655L697 -46Z"/></g><g transform="matrix(.012,0,0,-0.012,49.595,-8.44)"><path id="g50-110" d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z"/></g><g transform="matrix(.012,0,0,-0.012,49.595,4.995)"><path id="g50-106" d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"/></g><g transform="matrix(.012,0,0,-0.012,52.958,4.995)"><path id="g54-34" d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z"/></g><g transform="matrix(.012,0,0,-0.012,60.298,4.995)"><path id="g58-47" d="M389 0V32C297 37 291 46 291 118V635C234 612 176 595 109 583V556L161 554C203 551 207 548 207 497V118C207 46 201 37 110 32V0H389Z"/></g><g transform="matrix(.017,0,0,-0.017,69.744,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"/></g><g transform="matrix(.012,0,0,-0.012,81.802,-7.578)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g><g transform="matrix(.012,0,0,-0.012,83.266,4.995)"><use xlink:href="#g50-106"/></g><g transform="matrix(.017,0,0,-0.017,89.826,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.012,0,0,-0.012,102.953,-7.578)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g><g transform="matrix(.012,0,0,-0.012,110.293,-7.578)"><path id="g50-226" d="M501 517C491 587 426 710 310 710C240 710 175 662 175 601C175 561 201 517 250 452C219 438 187 423 157 404C88 360 24 280 24 178C24 71 90 -12 195 -12C250 -12 297 6 334 32C412 87 452 170 452 249C452 335 408 396 333 481C259 565 229 601 229 621C229 642 245 652 275 652C359 652 427 583 490 499L501 517ZM364 233C364 142 323 33 222 33C173 33 119 81 119 178C119 265 156 335 193 374C213 395 242 413 272 425C308 384 364 319 364 233Z"/></g><g transform="matrix(.009,0,0,-0.009,116.299,-4.806)"><path id="g176-106" d="M256 606C256 635 239 657 207 657C171 657 147 619 147 593C147 563 173 543 194 543C234 543 256 573 256 606ZM233 98L216 125C191 101 156 75 139 75C133 75 132 82 139 107L214 377C226 420 224 454 198 454C169 454 98 415 31 351L46 323C81 353 110 372 116 372C129 372 126 356 121 338L57 101C31 5 47 -12 72 -12C115 -12 197 52 233 98Z"/></g><g transform="matrix(.017,0,0,-0.017,120.136,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.012,0,0,-0.012,133.658,4.134)"><use xlink:href="#g50-106"/></g><g transform="matrix(.017,0,0,-0.017,142.486,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.017,0,0,-0.017,157.352,0)"><path id="g113-74" d="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z"/></g></svg>

Al-Dubiban, Asmaa M.

doi:https://doi.org/10.1155/2015/473965

Abstract and Applied Analysis

On this page

Abstract Introduction Conclusion References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 473965 | https://doi.org/10.1155/2015/473965

Positive Definite Solutions of the Matrix Equation

Asmaa M. Al-Dubiban¹

Academic Editor: Ivan Ivanov

Received28 Mar 2015

Accepted15 Jun 2015

Published12 Jul 2015

Abstract

We investigate the nonlinear matrix equation , where is a positive integer and . We establish necessary and sufficient conditions for the existence of positive definite solutions of this equation. A sufficient condition for the equation to have a unique positive definite solution is established. An iterative algorithm is provided to compute the positive definite solutions for the equation and error estimate. Finally, some numerical examples are given to show the effectiveness and convergence of this algorithm.

1. Introduction

We consider the nonlinear matrix equation:where is an identity matrix, is a positive integer, are complex matrices, and , for . This type of equations arises in solving a large-scale system of linear equations. In many physical applications, we must solve the system of linear equationwhere arises from a finite difference approximation to an elliptic partial differential equation [1]. For example, letWe can rewrite as , whereWe can decompose the matrix to the decompositionif and only if is a solution of (1). In addition, similar types of (1) have many applications in various areas, including control systems, ladder networks, dynamic programming, control theory, stochastic filtering, statistics, and optimal interpolation problem [2–5].

During the last few years, many researchers worked to develop the theory and numerical approaches for positive definite solutions to the nonlinear matrix equations of the form (1) [6–9]. Recently, Duan et al. [10] showed that the equation always has a unique positive definite solution by using the fixed point theorem of mixed monotone operators. They proposed an iterative method for computing the unique positive definite solution. Lim [11] presented a new proof for the existence and uniqueness of the positive definite solution for the last equation. Yin and Liu [12] derived necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equations with ; they also proposed iterative methods for obtaining positive definite solutions of these equations. He and Long [13] obtained some conditions for the existence of positive definite solution of the equation . They also presented two iterative algorithms to find the maximal positive definite solution of this equation. Duan et al. [14] used the Thompson metric to prove that the matrix equation always has a unique positive definite solution and they gave a precise perturbation bound for the unique positive definite solution. Similar kinds of equations have been investigated by many authors [15–20].

In this paper, we study the positive definite solutions of (1). The paper is organized as follows. In Section 2, we derive necessary and sufficient conditions for the existence of positive definite solutions of (1). We give a sufficient condition for the existence of a unique positive definite solution of this equation. In Section 3, we propose an iterative algorithm to obtain the positive definite solutions of (1). Moreover, we investigate convergence rate of the proposed algorithm. Finally, we give some numerical examples in Section 4 to ensure the performance and the effectiveness of the suggested algorithm.

We start with some notations which will be used in this paper. The symbol denotes the set of complex matrices. denotes the complex conjugate transpose of . We write , if matrix is positive definite (positive semidefinite). If is positive definite (positive semidefinite), then we write . Moreover, we denote by the spectral radius of . , , and denote the spectral, infinity, and Frobenius norm. For and a matrix , is a Kronecker product. vec() is a vector defined by .

2. Conditions for the Existence of the Solutions

In this section, we will derive the necessary and sufficient conditions for existence of the positive definite solutions of (1). We give a sufficient condition for the existence of a unique positive definite solution of (1). We begin with some lemmas, which will play a key role to establish our results.

Lemma 1 (see [21]). If or , then or , for all , and or , for all .

Lemma 2 (see [22]). If and are matrices, then and .

Lemma 3 (see [23]). Let and be positive operators such that and ; then, , for every nonnegative operator monotone function on , where if and if . Here, stands for a unitarily invariant norm on matrices.

Theorem 4. If (1) has a positive definite solution , then

Proof. Since is a positive definite solution of (1), thenby Lemma 1, we have . Also, from inequality and Lemma 1, we have . Therefore,Hence, implies that

Theorem 5. Equation (1) has a positive definite solution if and only if the coefficient matrices , , can be factored aswhere is a nonsingular matrix, , and is column-orthonormal. In this case, is a solution of (1).

Proof. Let be a positive definite solution of (1); then, , where is a nonsingular matrix. Furthermore, (1) can be rewritten aswhich implies thatLet ; then, and (11) turns intothat is,which means that is column-orthonormal.
Conversely, suppose that has the decomposition (9). Let ; then, is a positive definite matrix, and we haveHence, is a positive definite solution of (1).

Theorem 6. If are invertible, then (1) has a positive definite solution if and only if , , can be factored aswhere is unitary, are diagonal matrices such that , and is column-orthonormal. In this case, is a solution of (1).

Proof. Let be a positive definite solution of (1); then, by spectral decomposition of , we have , where is unitary and is diagonal. Therefore, (1) can be rewritten aswhich implies thatLet and . Then, , , is a diagonal matrix and . Moreover, (18) turns into , which means that is column-orthonormal.
Conversely, suppose that has the decomposition (15). Let ; then is a positive definite matrix, and we have Hence, is a positive definite solution of (1).

Theorem 7. If , then (1) has a unique positive definite solution.

Proof. We consider the mapping .
Let . Obviously, is a bounded closed convex set and is continuous on .
If , then by Lemma 1, andthat is,which means that . By Brouwer’s fixed point theorem, has a fixed point in ; thus, (1) has a solution in .
Now, we prove that is a contraction mapping on . For arbitrary , we getAccording to the definition of the mapping , we haveFrom (22) and (23), and by Lemmas 2 and 3, we haveSince , then is a contraction mapping on . By Banach’s fixed point theorem, the mapping has a unique fixed point in . This means that (1) has a unique positive definite solution in .

3. The Iterative Algorithm

In this section, we consider the fixed point iteration method and its speed of convergence. We present the following iterative algorithm to compute the positive definite solution of (1).

Algorithm 8. Consider

Theorem 9. If , then (1) has a unique positive definite solution and satisfieswhere the sequence , , is determined by Algorithm 8.

Proof. According to Theorem 7, (1) has a unique positive definite solution. Consider the sequence generated from Algorithm 8 and using Lemma 1.
Since , then we havewhich implies thatthen,that is,We find the relation between , , , and . Using , we obtainHence, . By the same way, we can prove thatSo, assume that the inequalitieshold for .
Now, for and using inequalities (34), we haveSimilarlyTherefore, inequalities (34) are true, for all ; that is, the subsequences and are monotonic and bounded. Hence, these are convergent to positive definite matrices.
To prove the subsequences , have a common limit, we haveSince , for each , and , then . By using Lemma 3, we haveSincefrom (38) and (39), we have Hence,that is,Consequently, the subsequences and are convergent and have a common limit , which is a positive definite solution of (1).

From Theorem 9, we can deduce the following corollary.

Corollary 10. From inequality (27), one has the following upper bound:

Theorem 11. If (1) has a positive definite solution and after iterative steps of Algorithm 8 one has , then

Proof. By using Algorithm 8, we have Taking norm of the above equation, we get By Theorem 9, we have , , and . By using Lemma 3, we have

4. Numerical Examples

In this section, we give some numerical examples to illustrate the convergence of the proposed algorithm. The solution is computed for different matrices , , with different orders and different values of and , . We denote , the solution obtained by Algorithm 8 and , , and .

For computing with an integer, we use the following iterative algorithm.

Algorithm 12 (see [24]). Consider

Example 1. Consider the matrix equationwhere , , and are given byBy using Algorithms 8 and 12, we have after 16 iterationsThe other results are listed in Table 1.

Example 2. Consider the matrix equationwhere , , and are given byBy using Algorithms 8 and 12, we have after 13 iterationsThe other results are listed in Table 2.

Example 3. Consider the matrix equationwhere , , , and are given byBy using Algorithms 8 and 12, we have after 18 iterationsThe other results are listed in Table 3.

Example 4. Consider the matrix equationwhere and are given byBy using Algorithms 8 and 12, we have solutions when . The results are listed in Table 4.

5. Conclusion

We have presented existence theorems of positive definite solutions for a nonlinear matrix equation (1) under certain conditions. Uniqueness theorem of solutions for this matrix equation is proved. An iterative algorithm is introduced to obtain positive definite solutions for this matrix equation. Moreover, convergence and error analysis are studied. Finally, the numerical examples confirmed the efficient convergence and error reduction as predicted and this method is efficient and easy.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson's equations,” SIAM Journal on Numerical Analysis, vol. 7, no. 4, pp. 627–656, 1970.
View at: Publisher Site | Google Scholar | MathSciNet
W. N. Anderson Jr., T. D. Morley, and G. E. Trapp, “Positive solutions to $X = A - {B X}^{- 1} B^{*}$ ,” Linear Algebra and Its Applications, vol. 134, pp. 53–62, 1990.
View at: Publisher Site | Google Scholar
L. A. Sakhnovich, Interpolation Theory and Its Applications, vol. 428 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
View at: Publisher Site | MathSciNet
J. Zabczyk, “Remarks on the control of discrete time distributed parameter systems,” SIAM Journal on Control, vol. 12, pp. 721–735, 1974.
View at: Publisher Site | Google Scholar | MathSciNet
X. Zhan, “Computing the extremal positive definite solutions of a matrix equation,” SIAM Journal on Scientific Computing, vol. 17, no. 5, pp. 1167–1174, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. M. El-Sayed and A. C. M. Ran, “On an iteration method for solving a class of nonlinear matrix equations,” SIAM Journal on Matrix Analysis and Applications, vol. 23, no. 3, pp. 632–645, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
S. M. El-Sayed, “Two iteration processes for computing positive definite solutions of the equation $X - A^{*} X^{- n} A = Q$ ,” Computers and Mathematics with Applications, vol. 41, no. 5-6, pp. 579–588, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
I. G. Ivanov, V. I. Hasanov, and F. Uhlig, “Improved methods and starting values to solve the matrix equations $X \pm A^{*} X^{- 1} A = I$ iteratively,” Mathematics of Computation, vol. 74, no. 249, pp. 263–278, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
X.-G. Liu and H. Gao, “On the positive definite solutions of the matrix equations $X^{s} \pm A^{T} X^{- t} A = I_{n}$ ,” Linear Algebra and Its Applications, vol. 368, pp. 83–97, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X - \sum_{i = 1}^{m} A_{i}^{*} {X^{δ_{i}} A}_{i} = Q$ ,” Linear Algebra and Its Applications, vol. 429, no. 1, pp. 110–121, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Lim, “Solving the nonlinear matrix equation $X = Q + \sum_{i = 1}^{m} M_{i} X^{δ_{i}} M_{i}^{*}$ via a contraction principle,” Linear Algebra and Its Applications, vol. 430, no. 4, pp. 1380–1383, 2009.
View at: Publisher Site | Google Scholar
X. Y. Yin and S. Y. Liu, “Positive definite solutions of the matrix equations $X \pm A^{*} X^{- q} A = Q (q \geq 1)$ ,” Computers and Mathematics with Applications, vol. 59, no. 12, pp. 3727–3739, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-M. He and J.-H. Long, “On the Hermitian positive definite solution of the nonlinear matrix equation $X + \sum_{i = 1}^{m} A_{i}^{*} X^{- 1} A_{i} = I$ ,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3480–3485, 2010.
View at: Publisher Site | Google Scholar
X. Duan, Q. Wang, and A. Liao, “On the matrix equation $X - \sum_{i = 1}^{m} N_{i}^{*} X^{- 1} N_{i} = I$ arising in an interpolation problem,” Linear and Multilinear Algebra, vol. 61, pp. 1192–1205, 2013.
View at: Google Scholar
X. Duan and A. Liao, “On hermitian positive definite solution of the matrix equation $X - \sum_{i = 1}^{m} A_{i}^{*} X^{r} A_{i} = Q$ ,” Journal of Computational and Applied Mathematics, vol. 229, pp. 27–36, 2009.
View at: Google Scholar
X. Duan, C. Li, and A. Liao, “Solutions and perturbation analysis for the nonlinear matrix equation $X + \sum_{i = 1}^{m} A_{i}^{*} X^{- 1} A_{i} = I$ ,” Applied Mathematics and Computation, vol. 218, pp. 4458–4466, 2011.
View at: Google Scholar
D. Gao, “Existence and uniqueness of the positive definite solution for the matrix equation $X = Q + A^{*} {(\hat{X} - C)}^{- 1} A$ ,” Abstract and Applied Analysis, vol. 2013, Article ID 216035, 4 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
V. I. Hasanov, “Notes on two perturbation estimates of the extreme solutions to the equations $X \pm A^{*} X^{- 1} A = Q$ ,” Applied Mathematics and Computation, vol. 216, pp. 1355–1362, 2010.
View at: Google Scholar
S. Vaezzadeh, S. M. Vaezpour, R. Saadati, and C. Park, “The iterative methods for solving nonlinear matrix equation $X + A^{*} X^{- 1} A + B^{*} X^{- 1} B = Q$ ,” Advances in Difference Equations, vol. 2013, article 229, 10 pages, 2013.
View at: Publisher Site | Google Scholar
S. Vaezzadeh, S. M. Vaezpour, and R. Saadati, “On nonlinear matrix equations,” Applied Mathematics Letters, vol. 26, no. 9, pp. 919–923, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. Parodi, La Localisation des Valeurs Caractéristiques des Matrices et ses Applications, Gauthier-Villars, Paris, France, 1959.
H. Lütkepohl, Handbook of Matrices, John Wiley & Sons, London, UK, 1996.
View at: MathSciNet
R. Bhatia, Matrix Analysis, vol. 169 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 1997.
View at: Publisher Site | MathSciNet
C.-H. Guo and N. J. Higham, “A Schur-Newton method for the matrix p'th root and its inverse,” SIAM Journal on Matrix Analysis and Applications, vol. 28, pp. 788–804, 2006.
View at: Google Scholar

Copyright

Copyright © 2015 Asmaa M. Al-Dubiban. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

2089

Downloads

1080

Citations