RESEARCH ON THE CALCULATION METHODS AND APPLICATIONS OF HIGH-POWER MATRICES

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 37 RESEARCH ON THE CALCULATION METHODS AND APPLICATIONS OF HIGH-POWER MATRICES Wang Yan JiNing Normal University, School of Mathematics and statistic, Ulanqab, Inner Mongolia, P. R. China https://doi.org/10.5281/zenodo.18041935 Abstract. The calculation of high-power matrices is one of the core issues in advanced algebra theory and its applications, and their efficient solution is of great value to the fields of science and engineering. This paper systematically combs seven general solution methods for high-power matrices, including: mathematical induction, binomial theorem, similarity diagonalization method, degree reduction method, matrix decomposition method, elementary transformation method, and block matrix method. Through specific case analyses, the applicable conditions and limitations of each method are summarized. By integrating these calculation methods, this paper constructs a methodological framework for the calculation of high-power matrices, aiming to provide systematic reference and guidance for relevant theoretical research and practical applications. Keywords: high-power matrix; calculation method; similarity diagonalization; Jordan canonical form; Cayley-Hamilton theorem. 1 Introduction The calculation of high-power matrices is a fundamental yet core problem in advanced algebra, playing a crucial role in solving differential equations, Markov chain analysis, graph theory, and various scientific computing and engineering applications. Accurately and efficiently computing high powers of matrices remains a significant challenge in both theoretical research and practical applications. To address this issue, a systematic methodology has been developed, ranging from basic mathematical induction to complex matrix decomposition techniques. This paper aims to systematically integrate and elaborate on the commonly used calculation methods for high-power matrices. It focuses on seven typical approaches: mathematical induction, the binomial theorem, the similarity diagonalization method, the degree reduction method, matrix decomposition methods (Jordan canonical form and the associative law of multiplication), the elementary transformation method, and the block matrix method. By combining specific computational examples, the principle, applicability, and limitations of each method are analyzed, ultimately constructing a methodological framework for calculating high-power matrices. 2 Calculation Methods for High-Power Matrices 2.1 Mathematical Induction Mathematical induction is a method that identifies patterns by calculating low-power matrices and proves the general term formula through mathematical induction, which can reduce redundant calculation steps. Specifically, if a recurrence relation   12 , , , k k k A f A A A   exists, the known low-power matrices can be used to quickly derive high-power matrices. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 38 Example 1: Let 1 0 0 1 0 1 0 1 0 A      , prove that when 3n , 22nn A A A E     is always true, and find 100 A . Solution: Mathematical induction is adopted. When 3n , we have 2 1 0 0 1 1 0 1 0 1 A      , 32 1 0 0 2 0 1 110 A A A E          , which means 32 A A A E   holds. Assume that the conclusion holds when nk , i.e., 22kk A A A E     . Then, when 1nk , we have   1 2 2 1 3k k k k A A A A A E A A A A              1 2 1 2kk A A A E A A A E          . Thus, by mathematical induction, when 3n , 22nn A A A E     is always true. Using the proven formula, we obtain   100 98 2 96 2 2A A A E A A E        2 2 2 49 50 49A A E A E      1 0 0 50 1 0 50 0 1      . The advantage of mathematical induction lies in its intuition and ease of understanding, making it suitable for low-order matrices or matrices with obvious patterns. However, it also has limitations: specifically, it requires proving the recurrence formula through mathematical induction, and is ineffective for matrices without regular patterns. 2.2 Binomial Expansion Method If matrix A can be decomposed into A F G , with the requirements that the powers of matrices F and G are easy to calculate, and FG GF (i.e., F and G are commutative), then by expanding via the binomial theorem, we obtain   1 1 2 2 1 1 0 n n n i n i i n n n n n n n n n n i A F G C F G F C F G C F G C FG G                 . This method is suitable for calculating the high powers of triangular matrices. In the calculation process, it is only necessary to directly take the powers of the main diagonal elements, while the non-diagonal elements are computed via the binomial theorem. For example, decompose the upper triangular matrix A into A E B   (where B is a nilpotent matrix, i.e., there exists m such that m BO ), then         1 0 n n n n n k n n k k n k A E B E n E B B C E B               . SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 39 The calculation of the high powers of a lower triangular matrix is similar to that of an upper triangular matrix, except that the recurrence direction of the non-diagonal elements is opposite. Example 2: Given matrix 10 01 00 A         , find k A . Solution: Decompose matrix A into 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 EHA                            , where 0 1 0 0 0 1 000 H      , and matrix H satisfies 2 3 4 0 0 1 0 0 0 , 000 H H H O          . Furthermore,     E H H EH      , i.e., E  and H commute. Thus, by the binomial expansion formula, we obtain         12 1 2 2 k k k k kkk A C H CEEHH E E            1 2 2 1 2 k k k kk E k H H         12 1 0 00 1 2 k k k kk k kk k k                  . The matrix in this problem is an upper triangular matrix, where non-zero elements lie on and above the main diagonal, and its powers still maintain the upper triangular structure. For a kth order strictly upper triangular matrix N (with a nilpotency index of k), the expansion of its high powers contains at most k terms, which significantly reduces the computational complexity. Decompose the matrix into A F G ; typically, matrix F is a diagonal matrix (or a scalar matrix E  ) composed of the diagonal elements of matrix A, and matrix G A F is a strictly upper triangular or lower triangular matrix. Using m BO to truncate m A into a finite number of terms reduces the computational complexity compared with general matrix expansion, but its computational efficiency is limited for high-order matrices. The advantage of the method using the binomial theorem is that it avoids direct matrix multiplication, making it suitable for matrices with identical diagonal elements, matrices SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 40 containing nilpotent components, block matrices, and special symmetric matrices. However, it also has limitations: it relies on the commutativity of submatrices and has low universality. 2.3 Similar Diagonalization Method If there exists an invertible matrix P for matrix A such that 112 (,,), n P AP diag     , i.e., A is similar to the diagonal matrix 12 ), , ,( n diag     , then 1 A P P  , and thus 1nn APP  . Example 3: Given matrix 3 1 1 1 2 0 1 0 2 A      , find matrix k A . Solution:     3 1 1 1 2 0 1 2 4 1 0 2 EA                    , Thus, the eigenvalues of matrix A are 1 2 3 1, 2, 4       . The corresponding eigenvectors can be obtained as       1 2 3 1,1,1 , 0, 1,1 , 2,1,1 T T T p p p     , let 1 0 2 1 1 1 1 1 1 P        , then 1 100 0 2 0 0 0 4 P AP          . Therefore 1 1 0 2 1 0 0 2 2 2 1 1 1 1 0 2 0 0 3 3 6 1 1 1 0 0 4 2 1 1 k k k k A P P                                2 2 2 1 2 1 2 1 2 2 2 1 2 2 2 2 2 2 2 2 12 2 2 3 2 2 2 3 2 2 62 2 2 3 2 2 2 3 2 2 k k k k k k k k k k k k k                                 . This problem mainly involves diagonalizing matrix A by solving its eigenvalues and eigenvectors, thereby calculating the matrix power. Although using diagonal matrices to compute high powers of matrices is a general method, it requires not only finding the eigenvalues of the matrix but also its eigenvectors, which remains relatively difficult for high-order matrices. The advantage of the similar diagonalization method lies in its high efficiency and speed, allowing for one decomposition and multiple uses. It is suitable for diagonalizable matrices, SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 41 diagonal matrices, and symmetric matrices. However, it also has limitations: it requires the matrix to be diagonalizable, resulting in low universality. 2.4 Reduction of Order Method Cayley-Hamilton Theorem: Let A be an nn matrix over the number field P, and () Af E   be the characteristic polynomial of A; then       1 11 22 10 n nn nn f A A a a a A A E           . To compute the high powers of a matrix using the reduction of order method, let 1 1 1 0 () nn nn f a a a a           , 00a be the characteristic polynomial of square matrix A; then       nf q r      , where   1 0 1 1 n n r r r r          can be obtained by the polynomial division with remainder. When  is an eigenvalue of A, we have =( 0)f  , i.e.,   nr   . Substituting A for  , we obtain       n A f A q A r A , According to the Cayley-Hamilton Theorem, we have () 0fA ; thus   1 0 1 1 nn n A r A r E rA r A        , According to this formula, n A can be obtained by determining the undetermined coefficients 0 r , 1 r ,…, 1n r . Example 4: Let 1 0 0 2 1 0 1 2 1 A       , find 100 A . Solution: The characteristic polynomial of A is     2 00 ( ) 2 0 12 1 1 1 1 1 fEA                . Since () 0f   , the eigenvalues are obtained as 12 1   , 31   . Through the polynomial division with remainder, we get     100 2 f q a b c          , (1) Differentiating the above formula, we obtain         99 100 2f q f q a b              , (2) and     1 1 0ff   ,   10f . Substitute 1   , 1   into Equation (1), and then substitute 1   into Equation (2) to obtain the system of equations SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 42 1 1 2 100 abc a b c ab           , solve the system to get 50 0 49 a b c       . Therefore     2 50 49 nfq        , and thus 2 1 0 0 1 0 0 50 49 50 0 1 0 49 0 1 0 6 0 1 0 0 1 n A A E                         1 0 0 0 1 0 300 0 1      . The matrix given in this problem is a lower triangular matrix, which can be computed using the binomial expansion method, but it still requires calculating matrix powers. In contrast, the reduction of order method avoids direct matrix multiplication and eigenvector computation. It only needs to derive the characteristic polynomial, and then use () 0fA to perform the calculation through undetermined coefficients. If the eigenvalues of the given matrix are distinct, it is sufficient to substitute each eigenvalue into       nf q r      and solve the linear system of equations; if there are repeated eigenvalues, it is necessary to construct an additional equation using differentiation to form a linear system. The advantages of the reduction of order method lie in avoiding direct multiplication and eigenvector computation, making it suitable for orthogonal matrices and low-dimensional triangular matrices. However, it also has limitations: it requires solving linear systems of equations, which is prone to calculation errors. 2.5 Matrix Decomposition Method For certain special matrices A, they can be decomposed into the product of two or more relatively simple matrices, thereby calculating n A ; this method is called the matrix decomposition method. Below, two main decomposition approaches are introduced: the matrix decomposition method using the Jordan canonical form and the one utilizing the associative law of multiplication. 2.5.1 Using the Jordan Canonical Form Any matrix over the complex number field is similar to a Jordan canonical form, which is a block diagonal matrix. Therefore, for a non-diagonalizable matrix A, n A can be obtained by calculating the power of its Jordan canonical form. In the computation, first find the Jordan canonical form J of matrix A and an invertible matrix P such that 1 P AP J  ; then 1kk A PJ P  . SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 43 Example 5: Given matrix 51 13 A    , find 100 A . Solution: The characteristic equation of matrix A is 2 det( ) 8 16 0AE         , A double root 4   is obtained by solving the equation. Substituting the eigenvalue 4   into Equation   40A E v , we get 11 00 11 xxy y               , so the eigenvector is 1 1 1 v    . Since there is only one linearly independent eigenvector, matrix A is non-diagonalizable. Construct a generalized eigenvector. For 4   , solve the equation   1 4A E v v , i.e. 1 1 1 1 1 1 x y                    , we obtain 2 0 1 v   . Let the invertible matrix be 10 11 P    , then 110 11 P   , therefore, the Jordan canonical form of matrix A is 41 04 J   , so 100 99 100 100 4 100 4 04 J    . Since 1 A PJP  , 100 99 100 100 1 100 1 0 1 0 4 100 4 1 1 1 1 04 A PJ P                  99 104 100 4100 104      . SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 44 In this problem, matrix A is neither diagonalizable nor a triangular matrix. Since any matrix over the complex field is similar to its Jordan canonical form, and the Jordan canonical form is a triangular matrix, calculations can be performed using 1 P AP J  . The matrix decomposition method based on the Jordan canonical form is applicable to all matrices; however, if a matrix is diagonalizable, the similarity diagonalization method is preferred. The matrix decomposition method using the Jordan canonical form has strong universality and is suitable for non-diagonalizable matrices. Nevertheless, it also has limitations: the calculation of generalized eigenvectors is highly complex, and its theoretical value outweighs its practical value due to the complexity of computations. 2.5.2 Utilization of the Associative Law of Multiplication If condition   1rA is satisfied, then matrix A can be decomposed into the product of a column vector and a row vector, i.e., T A   , and thus      2T T T T A        , let T l   , given 2T A l lA   , 3 2 2 A A A A lA lA      2 lA , then       11 11 nn n n T T T T T T n T n A l l A tr A A                     , where   Tii l tr A a      denotes the sum of the diagonal elements of the matrix (i.e., the trace of the matrix). Specifically, if 0l holds, then 1nn A l A   ; if 0l (where A is a nilpotent matrix), then 0 n A when 2n . If the rows and columns of a matrix are proportional to each other, the matrix can be decomposed into the product of a column vector and a row vector. In this case, the power of the matrix can be calculated using the aforementioned method, which simplifies the computation significantly. Example 6: Given 2 6 4 1 3 2 2 6 4 A          , find n A . Solution: Since 2 6 4 1 3 2 1 3 2 0 0 0 2 6 4 0 0 0 A                          , we have   1RA . Let   2, 1,2 T   ,   1,3,2 T   ,then T A   .Thus     n copies of matri 1 xT nn n T T T T T T A              , SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 45   2 1,3,2 1 2 3 4 3 2 T              . and therefore   11 3 n n T T n T A        11 2 6 4 123 3 2 6 4 3 nn A            . In this problem, the first row and the third row of matrix A are identical, and the second row is 1 2  times the first row. It follows that the rank of matrix A is 1. Therefore, matrix A can be transformed into the product of an n-dimensional column vector   12 ,, T n a a a   and an ndimensional row vector   12 ,, Tn b b b   , i.e., T A   . When calculating high powers of the matrix, the associative law of multiplication is applied. Since T  is a constant,       1 1 1n n n n T T T T T AA                . This method avoids complex matrix multiplication in the computation. The matrix decomposition method utilizing the associative law of multiplication offers advantages such as low computational complexity and a concise, intuitive form. Without the need for complex decomposition, the calculation can be directly completed by multiplying the power of the inner product with the original matrix. This method is applicable to all matrices of rank 1. However, it has limitations due to its strict reliance on the rank-1 condition—if the rank of the matrix is greater than 1, the method cannot be directly applied. 2.6 Elementary Transformation Method The elementary matrix decomposition method simplifies computations by expressing matrix A as the product of high powers of elementary matrices, leveraging the power properties of elementary matrices. Elementary matrices are classified into three categories: (1) Interchange Matrix ij E : Swap the i-th row and j-th row of the matrix, which satisfies 2 ij EE (where E is the identity matrix). It has periodicity, and thus , is an even number , is an odd number n ij ij En EEn    ; (2) Scaling Matrix   i Ek : Multiply the i-th row of the matrix by a non-zero constant k, which satisfies     nn ii E k E k ; (3) Row Addition Matrix   ij Ek : Add k times the j-th row of the matrix to the i-th row, which satisfies     n ij ij E k E n k . If matrix A can be decomposed into the product of powers of elementary