We discuss an algorithm with a simplistic approach to solving systems of linear equations arising from the application of real-valued vector space ideas to the computation of the large powers of square matrices

Dogan, Hamide

English

Hamide Dogan

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Vector Space Ideas Computing A^m

Scitech Research Journals

Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218  Volume 11, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                   1530|                  SCITECH                                                                         Volume 11, Issue 1  RESEARCH ORGANISATION|                 Published online: January 14, 2017|        Journal of Progressive Research in Mathematics   www.scitecresearch.com/journals     Vector Space Ideas Computing Am  Hamide Dogan UTEP Mathematics EL Paso, TX United States  Abstract.  We discuss an algorithm with a simplistic approach to solving systems of linear equations arising from the application of real-valued vector space ideas to the computation of the large powers of square matrices.  Keywords: Matrix Power Algorithm; Linear Algebra; Vector Spaces; Linear Systems.  1. Introduction Large powers of matrices are commonly applied to many areas such as graph theory models and population prediction models etc. So far, the calculation of powers of matrices however requires computationally taxable algorithms that need and require necessary number of eigenvalues of matrices for diagonalization [4], or algorithms that need to apply computer-based memory-consuming recursive steps [2,3]. In this article, we derive a simple approach of solving systems of linear equations in order to obtain the large powers of square matrices with very little memory usage. Our approach (we named “Augmented Approach”) makes use of the division algorithm over the ring of polynomials [1], and it does not require any recursive process or the computation of the eigenvalues of matrices. 2. Vector Space Ideas  Using the vector space structure of the set of polynomials of degree m, Pm, over the field of Real numbers, one writes any polynomial of degree m (or less), G(x)=mii xz0, applying the division algorithm using a basis of the vector space [1] as:   G(x)= mii xz010nii xa +nmii Fxb0  where F(x)= nii xy0for n≤m (1) Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218  Volume 11, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                   1531|                  Here, {1, x, x2, …,xn-1, F, xF, x2F,…,xm-nF} is the basis of Pm, used in (1), which guarantees a unique linear combination of the basis vectors resulting in G(x) [1]. Solving equation (1) for the unknowns bi and ai leads to the recognition of two separate systems of linear equations. One of the two systems contains equations with only the unknowns bi, and the other includes equations with each having single unknown, ai.  Equations of the first system are obtained by comparing the coefficients of the terms xm-k. These coefficients form a ((m-n)+1)x((m-n)+2) system of linear equations(Here, we set yɟ=0 when ɟ<0): kmkiiniknm zyb  0, k=0,…,m-n   (2) It is easy to verify that (2) gives the coefficients of xm-k: iknmb   is the coefficient of Fxiknm   , and iny   is the coefficient of the terminx  in F. Thus, (2) picks the coefficient of the term Fx iknm  , and matches with the coefficient of inx   making sure the combined power of x is adding upto the power m-k. For instance, for the largest possible power m of x, equation in (2) considers k=0, and picks the coefficient nmb   of xm-nF with the highest power, and pairs it with the coefficient ny of xn in F with the highest power, which are the only combinations adding upto the largest power xm.  In short, (2) considers the coefficients of x ranging from n to m.  An interesting result emerges when considering the augmented matrix of the system in (2). One obtains a nicely patterned ((m-n)+1)x((m-n)+2) triangular matrix seen in (3): nmmmnmnnmnnnnnnnnzzzzyyyyyyyyy....................0....00...0021)(1)(211                                         (3)  In light of the triangular aspect of the matrix in (3), readers may agree that solving the system in (2) requires only the basic back substitution technique. Also it is easy to verify that this system has a unique solution whenever 0ny .  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218  Volume 11, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                   1532|                 After obtaining the values of the unknowns, bi, it is not hard to evaluate the values of ai. All one needs is to substitute bi values to the single unknown equations given in (4) below, and apply basic arithmetic. The values of ai are in turn forming the remainder polynomial in the division algorithm (1). It can easily be verified that the equations in (4) are obtained from comparing the coefficients of xi where the index i ranges from 0 to n-1:                         ii za  -ikkki yb0, i=0,…,n-1                                                                                (4) We should note here that equations in (2) could also be obtained from the expression in (4) by extending the values of the index i from n-1 tom, and adding the following two conditions: bt=0 if t>m-n & ys=0 if s>n. Example: Let us consider a polynomial F with degree 2, and a polynomial G with degree 5. Then,   F(x)=20iii xy and G(x)=50iii xz . Using the equation in (1), we get:       FxbFxbxFbFbxaaxzxzxzxzxzz 332210105544332210                        (5) Solving (5) via the comparison of the coefficients of xk, k=0, …,m, we get the values of the unknowns bi and ai.                Equations with unknowns ai: o Comparing the constant terms in (5) gives the equation:  0000 ybza  . This further verifies the equation in (4) for i = 0.  o Comparing the coefficients of the x-terms in (5) gives the equation: 011011 ybybza  ;  This verifies the equation in (4) for i=1. Equations with Unknowns bi: o Comparing the coefficients of the x2-terms in (5) gives the equation:  0211202 ybybybz   ;  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218  Volume 11, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                   1533|                 This time, the formula in (2) can be verified by setting k=m-n=3:   35302325   zybiii.Next, expanding out the summation gives: 213021120 zybybybyb   .   Finally, setting y-1=0, gives the linear equation: 2021120 zybybyb  . o Comparing the coefficients of the x3-terms in (5) gives the equation:  0312213 ybybybz  . Again, one can easily verify the formula in (2) by setting k=2. o Comparing the coefficients of the x4-terms in (5) gives the equation:  13224 ybybz  .  The same equation emerging from the expression in (2) can be verified by setting k=1. o Comparing the coefficients of the x5-terms in (5) gives the equation: 235 ybz  .  One can easily verify (2) setting k=0. Notice that the coefficients of xk, k=2,3,4,5 forms a 4x5 linear system containing only the unknowns, bi, i=0, 1,2,3. Furthermore, its augmented matrix is a 4x5 triangular matrix with a nice pattern as seen in (6). The pattern emerged in (6) is also in agreement with the matrix form given in (3).                                                 20123012412520000000zyyyzyyyzyyzy                                                                 (6) The triangular and easily recognized pattern in (6) would make it easier to input these matrices into computer algorithms, and furthermore it would be straightforward for the computer-based programs to solve the systems easily simply by applying basic back-substitution techniques thus using drastically less memory than memory used with many of the existing approaches such as eigenvalue approach to diagonalizing matrices [4], and recursive approach to obtaining remainder polynomials in division algorithms [2,3].  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218  Volume 11, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                   1534|                 3. Computing Am In fact, the pattern observed in the matrix (3) has implications for the calculation of higher powers of matrices. It has been known that the Division Algorithm and Cayley-Hamilton Theorem provide means to be able to calculate a large power of a square matrix by evaluating a remainder polynomial, obtained via division algorithm, for the matrix via a recursive process [2, 3]. In this case, F polynomial in the equation (1) takes the role of the characteristic polynomial of a square matrix, and the polynomial G considers only the xm-type polynomials. The recursive process however can be memory consuming for computer-based algorithms to compute large values of the powers of remainder polynomials. Our Augmented Approach drastically cuts down the processes of finding a remainder polynomial thus provides a more effective means to obtain a remainder polynomial and as a result more efficient calculation of Am for the large values of m. 3.1 Augmented Approach  Our approach gives the nicely pattern triangular augmented matrix below in (7) obtained by considering the characteristic polynomial, F, of an nxn matrix, and applying the division algorithm (1) to the characteristic polynomial and the polynomial G(x) = xm. Moreover, since we have 1ny  in the characteristic polynomial of any square matrix, the linear system represented by the matrix in (7) always has a unique solution.                                        0...001....1...........101....010...00)(1)(211nmnnmnnnnyyyyy                                                     (7)  Example: Let’s now calculate A6 using the Augmented Approach where A=400230012.  Its characteristic polynomial is F(x)=24 − 26𝑥 + 9𝑥2 − 𝑥3. Thus, for this example, the augmented matrix in (7) becomes a 4x5 triangular matrix seen in (8) representing the linear system obtained by applying the equation in (1): Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218  Volume 11, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                   1535|                                                       02426910269100910011000                                                                     (8)  The row reduced echelon form of the matrix in (8) or solving the system using back-substitution beginning with the top row in (8), -b3=1, gives the values:                                      19552853210bbbb Using these values, we easily evaluate the equations in (4) thus obtaining the remainder polynomial: R(x)=6840-6090x+1351x2 Now, we are ready to compute A6:   A6=R(A)= 6840I-6090A+1351A2 =40960067347290270266564. As a final remark regarding the higher powers of A in the example above, since y values are the same, the augmented matrix for higher powers will be a spare matrix with many zeros thus requiring about the same number of steps as the lower powers to calculate Am. As an example, for A100, the augmented matrix would be a 98x99 triangular matrix, seen in (9),with only non-zero entries coming from the values of yn of the characteristic polynomial Fof A. Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                 ISSN: 2395-0218  Volume 11, Issue 1 available at www.scitecresearch.com/journals/index.php/jprm                                                                   1536|                 0000...2691...............910002426...10...024269....000026910......00910000011000...00                                                   (9)  Notice that using our augmented matrix approach we directly obtained the coefficients of the remainder polynomial R for the calculation of A6 by simply solving a single linear system. That is, we did not need to compute the coefficients of the remainder polynomials for the earlier degrees, A3, A4,and A5. An already existing recursive approach derived from the division algorithm however requires the coefficients of the earlier remainder polynomials with lesser degrees calculated to be able to identify the coefficients of the remainder polynomial for the desired degree hence requiring a recursive process [2, 3].  References [1] A. Von Sohsten de Medeiros (2002) Elementary Linear Algebra and the Division Algorithm. The College Mathematics Journal. The Mathematical Association of America. [2] R. Abu-Saris and W. Ahmad(2005) Avoiding Eigenvalues in Computing Matrix Power, The American Mathematical Monthly , Vol. 112, Number 5, (May 2005) 450-454. [3] S.N. Elaydi and W. A. Harris, Jr. (1997) On the computation of An, SIAM Rev. 40 (1998) 965-971.  [4] A. Sheldon (1997). Linear Algebra Done Right. Second Edition. Springer.  

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Vector Space Ideas Computing A^m

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