A matrix x is a solvent of the matrix polynomials. f(x)=A o xm +…..+Am if f(x) = 0.Where A o, x, f are square matrices. In this paper we develop the linear algebra of matrix polynomials and solvents. we define division and interpolation the properties of matrices and define and study the existence of a complete set of solvents. we study the relation between the matrix polynomial problems and Theorems and lambda matrix problem, Which is to find scalar λ such that
A o λm+A1 λm-1+ ……. +Am is singular.
In a future paper we extend Cayley Hamilton Theorem for calculating of scalar polynomials to matrix polynomials. And lambda matrix for column and row Hermite calculating of lambda matrix

K, CHANDRAN

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1  MATRIX POLYNOMIAL AND LAMBDA MATRIX K.CHENDRAN1,Dr.R.G.BALAMURUGAN2,Dr.S.SANGEETHA3,R.RAMYA4 Department of Mathematics Dhanalakshmi Srinivasan College of  Arts and Science for Women (Autonomous) Perambalur  ABSTRACT: A matrix x is a solvent of the matrix polynomials.    f(x)=A o xm +…..+Am if f(x) = 0.Where A o, x, f are square matrices. In this paper we develop the linear algebra of matrix polynomials and solvents. we define division and interpolation the properties of matrices and define and study the existence of a complete set of solvents. we study the relation between the matrix polynomial problems and Theorems and lambda matrix problem, Which is to find scalar λ such that                    A o λm+A1 λm-1+ ……. +Am is singular.  In a future paper we extend Cayley Hamilton Theorem for calculating of scalar polynomials to matrix polynomials. And lambda matrix for column and row Hermite calculating of lambda matrix. KEYWORDS:                         Matrix polynomial, n square matrix, characteristic polynomial, scalar variable   INTRODUCTION  A polynomial in the scalar variable x having coefficients which are scalars is known as a scalar polynomial. But there are polynomials where the variables, instead of being a scalar is an n- square matrix 𝐴, but the coefficients remain scalars. Then, these polynomials are called matrix polynomials. On the other hand, if matrices take the place of the scalar coefficients in the original scalar polynomial, then we have a polynomial in the scalar variable x having coefficients which are matrices, and we call it a polynomial matrix (or) lambda matrix. These two 2  different types of polynomials involving matrices and their operational properties are the subject- matter of this section. DEFINITION 1.1:         The matrix polynomial of degree n given by |A-XI | is called the “Characteristic matrix” of A. DEFINITION 1.2:        The determinant | A-XI | which is ordinary polynomial in x of degree n over the field F is called the “Characteristic polynomial” of A.        The equation | A-XI |=0 is called characteristic equation of A. Definition 1.3:         The “Rank of matrix” A is common value of its row and column rank. Matrix polynomial and lambda matrix DEFINITION 2:1:     Considering x to be scalar variables the 𝑚𝑡ℎdegree polynomial in x denoted by f (x) is the functional relation. 𝑓 𝑥 =  𝑎𝑚𝑥𝑚 +  𝑎𝑚−1𝑥𝑚−1 + ⋯ +  𝑎1𝑥 +  𝑎0                          Where 𝑎𝑚  ≠ 0 is positive integer and the coefficient of𝑎0 is x0 we shall call f(x) as a monic polynomial if the leading coefficient 𝑎𝑚 = 1.                                                                                                                                            DEFINITION 2:2: An n– square matrix A (𝜆) which is a polynomial in the scalar variable λ from a field F represented by                       A (λ) = Am λ m + Am-1 λm-1 +……. + A1 λ +A0 Where the leading co efficient Am ≠ 0, Ai ‘s are square matrices in v n × n is a defined as a polynomial or a lambda matrix or simply a λ – matrix of order n and of degree m over a field F. 3             If the determinant of the leading coefficients is not zero. if det Am ≠ 0, Then the lambda matrix is said to be regular. THEOREM 2:1   Let two ʎ - matrices A (ʎ) and B (ʎ) of the same order with respective degrees m and p, m ≥ p, be given A (ʎ) and B (ʎ) with B (ʎ) regular. Then the right quotient Q (ʎ) and the right remainder R (ʎ) Of A (ʎ) on dividing by B (ʎ).                                     A (ʎ) = Q (ʎ) B (ʎ) + R (ʎ)  There exist and are Unique. Similar assertion hold for the left quotient and the left remainder.   PROOF:  To prove the existence, we shall proceed as we do in the case of long division of one polynomial by another lower or equal degree.so, we pre -multiply both side B(ʎ) by Am Bp-1 ʎ m-p to get                    𝐴 ʎ =  𝐴𝑚ʎ𝑚  +  𝐴𝑚−1ʎ𝑚−1 + ⋯ + 𝐴1 ʎ + 𝐴0……….. (1) 𝐵 ʎ =  𝐵𝑝ʎ𝑝  +  𝐵𝑝−1ʎ𝑝−1 + ⋯ + 𝐵1 ʎ + 𝐵0 ………… (2)       Am Bp-1 ʎ m-p𝐵 ʎ =  𝐴𝑚ʎ𝑚  +  𝐴𝑚𝐵𝑝− 1𝐵𝑝−1 ʎ𝑚−1 + ⋯ + Am Bp-1 B 0 ʎ m-p …….. (3) Subtraction (3) from (1) we get,                 A (ʎ) – Am Bp-1 ʎ m-p𝐵 ʎ = (Am-1 Am Bp-1𝐵𝑝−1) ʎ𝑚−1+…………… (4)        Since the first term on the right hand side (4) may be identically zero,     We get from (4),                     A (ʎ) - Am Bp-1 ʎ m-p𝐵 ʎ = A (1) (ʎ) ………….. (5) Where A (1) (ʎ) is an n-square ʎ - matrix with degree m1< m and represented by,                      A (1) (ʎ) = Am 1(1) ʎ m1 + …… + A o (1).     If m1 ≥ p, we repeat the above process on    A (1) (ʎ) rather than on A (ʎ) to get, 4                      A (1) (ʎ) - Am 1(1) Bp-1 ʎm1 - p𝐵 ʎ = 𝐴(2) (ʎ) ……… (6) Where,  𝐴(2)  (ʎ) = Am2 (2) ʎm2 + ………. +A o(2) .  Is again an n – square ʎ - matrix with degree m2 < m1. Since the degree of A (ʎ), A (1) (ʎ), 𝐴(2)  (ʎ) are the decreasing progressively, we finally arrive after a finite  number of (r + 1) steps at a polynomial A(r+1) (ʎ) with degree mr+1 < p satisfying                    A(r) (ʎ) - Amr(r) Bp-1 ʎ m r-p𝐵 ʎ =  A(r+1) (ʎ) ……….. (7) Where Amr(r) is the leading coefficients of A(r) (ʎ) with degree m r   ≥ p.  Adding (5) to (6) we get                      A (ʎ) – {[Am ʎ m + Am1 (1) ʎm1 + …. + Amr(r) ʎ m r] ʎ-p Bp-1} B (ʎ) = A(r+1) (ʎ),          Where it is not difficult to identify the expression within the braces with the right quotient Q (ʎ) and A(r+1) (ʎ) with the right remainder R (ʎ) in,                       A (ʎ) = Q (ʎ) B (ʎ) + R (ʎ) ………… (8) To prove, Uniqueness, We assume that Q1 (ʎ) is the another right Quotient and   R1 (ʎ) another right remainder so that,                       A (ʎ) = Q1 (ʎ) B (ʎ) + R1 (ʎ) ………… (9)  Subtracting (8) and (9), we obtain,                        [Q (ʎ) B (ʎ) + R (ʎ)] = [Q1 (ʎ) B (ʎ) + R1 (ʎ)]                        [Q (ʎ) B (ʎ) - Q1 (ʎ) B (ʎ)] = [R1 (ʎ) - R (ʎ)]                       [Q (ʎ) - Q1 (ʎ)] B (ʎ) = [R1 (ʎ) - R (ʎ)] …………. (10)  Where the degree of ʎ -matrix on the right hand side is the than that of B (ʎ). So if Q (ʎ) ≠ Q1 (ʎ).           We arrive at a contradiction in view of the Equality of two lambda matrices. Regarding the equality of two ʎ -matrix in (10)                                           Q (ʎ) =Q1 (ʎ). 5  Consequently, R1 (ʎ) = R (ʎ).             Similar proof can be given for the left quotient and left remainder. In general, the left quotient and the right quotient and the left remainder and right remainder are different.  THEOREM 2:2  REMANIDER THEOREM: Statement:    Suppose A (ʎ), an n – square ʎ - matrix of degree m described by,              𝐴 ʎ =  𝐴𝑚ʎ𝑚  +  𝐴𝑚−1ʎ𝑚−1 + ⋯ + 𝐴1 ʎ + 𝐴0………. (1)    And B an n – square matrix, are given. Then there exist a unique n – square         ʎ - matrix Q (ʎ) of degree m-1 represented by,                        Q (ʎ) = Qm-1 ʎm-1 + Qm-2 ʎm-2 + ……… + Q1 (ʎ) + Q0   ...........  (2)   And an n – square matrix R such that    𝐴 ʎ = Q (ʎ) (ʎ I – B) + R    ………………. (3)  Where R = A r (B) is the right hand value of A ( ʎ) 𝑎𝑡 B,                                   (OR) Where R= 𝐴r (B) is the right value of  𝐴 ʎ   at B,                                  (OR) Where R = 𝐴L (B) is the left value of  𝐴 ʎ   at B, and Q ( ʎ) is an n – square      ʎ - matrix of degree (m-1). PROOF: Subtracting in (3) equation the expressions for A (ʎ)and Q ( ʎ) from (1) and (2) respectively, we get  6  𝐴𝑚ʎ𝑚  + 𝐴𝑚−1ʎ𝑚−1 + ⋯ + 𝐴1 ʎ + 𝐴0 =                                                                  Qm-1 ʎm-1 + Qm-2 ʎm-2 +        ……… + Q1 (ʎ) + Q0   (ʎ I – B) + R.   In which equating coefficients of like powers of  ʎ  in view of definition   (ʎ𝐼 −A) X = θ for equality of two ʎ - matrices we have,                     Am = Qm-1 𝐴𝑚−1= - Qm-1 B + Qm-2   A1 = - Q1 B + Q ………… AO = - QO B + R    ……………… (4) Hence , since Ai ‘s and B are given, Qi ‘s can be determined uniquely from the first m equations .post multiplying equation (4) successively from below by I,B,B2…..Bm-1  B m  and  adding we get,  𝐴𝑚𝐵𝑚  + 𝐴𝑚−1𝐵𝑚−1 + ⋯ + 𝐴1 𝐵 + 𝐴0 = R Hence, R = A r (B).           Similarly, the left quotient Q (ʎ) can be uniquely determined with R= AL (B).   CONCLUSION:                        In the project briefly discussed matrix polynomial and λ matrix.  Also we have discussed some basic definition and some important derivates, Examples related theorem, and then we have discussed about the solution of the matrix polynomial equations, λ matrix equation and confluent matrix polynomials and λ matrix. BIBILOGRAPHY:   7  1) Gohberg, I Matrix polynomial Academic press, New York.1982 2) E.V. Krishnamurthy,error – free polynomial matrix computations New York 1985 3) Gilbert Stang (1988) linear Algebra and its application 2rd edition. 4) Stoll, R.R (1952), Linear Algebra and Matrix Theory. New York ;MC Graw Hill. 5) S. Axler, linear Algebra Done Right, Springer verlag .New York 1996 6) V. Sahai, and V. Bist, Algebra, Narosa Publishing House, 1999        

MATRIX POLYNOMIAL AND LAMBDA MATRIX

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MATRIX POLYNOMIAL AND LAMBDA MATRIX

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