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Quasi-Rational Canonical Forms of a Matrix Over a Number Field

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Quasi-Rational Canonical Forms of a Matrix Over a Number Field Advances in Linear Algebra & Matrix Theory, 2018, 8, 1-10 http://www.scirp.org/journal/alamt ISSN Online: 2165-3348 ISSN Print: 2165-333X Quasi-Rational Canonical Forms of a Matrix over a Number Field Zhudeng Wang*, Qing Wang, Nan Qin School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, China How to cite this paper: Wang, Z.D., Wang, Abstract Q. and Qin, N. (2018) Quasi-Rational Ca- nonical Forms of a Matrix over a Number A matrix is similar to Jordan canonical form over the complex field and the Field. Advances in Linear Algebra & Matrix rational canonical form over a number field, respectively. In this paper, we fur- Theory, 8, 1-10. ther study the rational canonical form of a matrix over any number field. We https://doi.org/10.4236/alamt.2018.81001 firstly discuss the elementary divisors of a matrix over a number field. Then, we Received: October 28, 2017 give the quasi-rational canonical forms of a matrix by combining Jordan and Accepted: January 7, 2018 the rational canonical forms. Finally, we show that a matrix is similar to its Published: January 10, 2018 quasi-rational canonical forms over a number field. Copyright © 2018 by authors and Keywords Scientific Research Publishing Inc. This work is licensed under the Creative Matrix, Jordan Canonical Form, Rational Canonical Form, Commons Attribution International Quasi-Rational Canonical Form License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access 1. Introduction A matrix is similar to Jordan canonical form over the complex field and the ra- tional canonical form over a number field, respectively. Thus, Jordan and the ra- tional canonical forms of a matrix over the complex field are similar. Recently, Radjabalipour [1] studied the symmetrization of the Jordan canoni- cal form, Abo et al. [2] and Barone et al. [3] discussed the relations between the eigenstructures and Jordan canonical form. Moreover, Li [4] discussed the prop- erty of the rational canonical form of a matrix, Liu [5] gave out a constructive proof of existence theorem for rational form, and Radjabalipour [6] investigated the rational canonical form via the splitting field. In this paper, we further study the rational canonical form over any number field. We firstly discuss the concept of elementary divisors of a matrix over any num- ber field. Then, we give the quasi-rational canonical forms of a matrix by combin- ing Jordan and the rational canonical forms. Finally, we show that a matrix is sim- ilar to its quasi-rational canonical forms over any number field. DOI: 10.4236/alamt.2018.81001 Jan. 10, 2018 1 Advances in Linear Algebra & Matrix Theory Z. D. Wang et al. 2. Jordan and Rational Canonical Forms Given a matrix A, it is an interesting work to find a simple matrix that is similar to A. We know that such a simple matrix is Jordan canonical form or the ration- al canonical form of A. Lemma 2.1. ([7], pp 244-247) An nn× matrix A is similar over the complex field to Jordan canonical form of its, such Jordan canonical form is unique up to a rearrangement of the order of its characteristic values, i.e., A is similar to the quasi-diagonal matrix of order n J1 J J = 2 Js where ci 0 00 1c 00 i Ji = 01 00 00 1ci mmii× is called an elementary Jordan matrix with characteristic value ci , is=1, 2, , and mm12+ ++ ms = n. nn−1 Definition 2.1. For a non-scalar monic polynomial daa(λλ) = +1 λ ++ n over a number field P, the nn× matrix 00 0 −an 10 0−a n−1 B = 01 0−an−2 00 1 −a1 is called the companion matrix or Frobenius matrix of the monic polynomial d (λ ) . A polynomial matrix, or λ-matrix, is a rectangular matrix A(λ ) whose ele- ments are polynomials in λ kk k−−11 k 0 A(λ) =( aij ( λ)) =( aaij λλ + ij ++ aij ) . Here k is the largest of the degrees of the polynomial aij (λ ) . Two polynomial matrices A(λ ) and B(λ ) are called equivalent if one of them can be obtained from the other by means of some elementary operations. An arbitrary rectangular polynomial matrix is equivalent to a canonical ma- trix a1 (λ ) 0 00 λ 0a2 ( ) 00 λ 00as ( ) 0 00 0 0 DOI: 10.4236/alamt.2018.81001 2 Advances in Linear Algebra & Matrix Theory Z. D. Wang et al. where the polynomials aa12(λλ),( ) ,, as ( λ) are not identically equal to zero and each is divisible by the preceding one. Let A(λ ) be a polynomial matrix of rank r, i.e., the matrix has minors of order r not identically equal to zero, but all the minors of order greater than r are identically equal to zero in λ. We denote by D j (λ ) the greatest common divi- sor of all the minors of order j in Aj(λ ) ( =1, 2, , r) . It is easy to see that in the series DD12(λλ),( ) ,, Dr ( λ) each polynomial is divisible by the preceding one (see [8], pp 139-140). An easy verification shows immediately that the elementary operations change neither the rank of A(λ ) nor the polynomials DD12(λλ),( ) ,, Dr ( λ) . Thus, rs= , the corresponding quotients will be denoted by DD2 (λλ) r ( ) d1(λ) = Dd 12( λλ),( ) = ,, dr (λ) = DD11(λλ) r− ( ) is invariant under elementary operations and d1(λ) = ad 12( λλ),( ) = a 2( λ),, drr( λ) = a( λ) . The polynomials dd12(λλ),( ) ,, dr ( λ) are called the invariant polynomials of the λ-matrix A(λ ) . Definition 2.2. ([8], pp 144-145) Let Aa= ( ij ) be an nn× matrix. We form its characteristic matrix λ −−aa11 12 − a1n −−aaλ − a λEA−=21 22 2n . −−aan12n λ − ann The characteristic matrix is a λ -matrix of rank n. Its invariant polynomials DD2 (λλ) n ( ) d1(λ) = Dd 12( λλ),( ) = ,, dn (λ) = DD11(λλ) n− ( ) are called the invariant polynomials of the matrix A. It is easy to see that the invariant polynomials of the companion matrix B of the monic polynomial d (λ ) are 1, , 1, d (λ ) . Definition 2.3. The following quasi-diagonal matrix B1 B C = 2 Bs is called the direct sum of the companion matrices Bi of non-scalar monic polyno- mials dd12(λλ),( ) ,, ds ( λ) such that di+1 (λ ) divides di (λ ) for is=1, , − 1 and said to be in rational canonical form. The invariant polynomials of the rational canonical form matrix B in Defini- tion 2.3 are DOI: 10.4236/alamt.2018.81001 3 Advances in Linear Algebra & Matrix Theory Z. D. Wang et al. 1, , 1,dd12(λλ) ,( ) , , ds ( λ) . Lemma 2.2. ([7], pp 238-239, Theorem 5) An nn× matrix A is similar over a number field P to one and only one matrix which is in rational canonical form, i.e., A is similar to the quasi-diagonal matrix B1 B C = 2 Bs where BB12,,, Bs are the companion matrices of the non-scalar invariant polynomials dd12(λλ),( ) ,, ds ( λ) of matrix A. 3. The Elementary Divisors of a Matrix over a Number Field Let P be a number field. Then a non-scalar monic polynomial in Px[ ] can be fac- tored as a product of monic irreducible polynomials in Px[ ] in one and, except for order, only one way. In the factorization of a given non-scalar monic polynomial fx( ) , some of the monic irreducible factors may be repeated. If px12( ), px( ) ,, pxs ( ) are the distinct monic irreducible polynomials occurring in this factorization of fx( ) , then rr12 rs fx( ) = p12( x) ⋅ p( x) ps ( x) the exponent ri being the number of times the irreducible polynomial pxi ( ) oc- curs in the factorization. This decomposition is also clearly unique, and is called the primary decomposition of fx( ) . Theorem 3.1. If B is the companion matrix of the monic polynomial nn−1 paa(λλ) = +1 λ ++ n B EB1n Cc= = ( ij ) rn× rn EB1n where 0 01 0 00 E = 1n 0 00nn× i.e., ccnn++1,= 2 n 1,2 n = = c(rn−+11,1) ( rn −) =1 then the invariant polynomials of C are 1, 1, , pr (λ ) . DOI: 10.4236/alamt.2018.81001 4 Advances in Linear Algebra & Matrix Theory Z. D. Wang et al. Proof. Let λEB− −−λ E1n EB C(λλ) = EC −= =(cij (λ)) . rn× rn −−E1n λ EB Then cc21(λλ) = 32 ( ) = = crn, rn− 1 ( λ) = −1 and a minor of order rn −1 cc21 (λλ) 22 ( ) c2,rn− 1 ( λ) cc31 (λλ) 32 ( ) c3,rn− 1 ( λ) = ±1. ccrn,1 (λλ) rn,2 ( ) crn, rn− 1 ( λ) Thus, D11(λ) = = Drn− ( λ) =1, Drn ( λλ) = EC − . By Laplace Theorem, we have that r r Drn (λλ) = EC −= λ EB − = p( λ) . Therefore, the invariant polynomials of C are 1, 1, , pr (λ ) . The theorem is proved. The matrix C is called the rational block of pr (λ ) and the characteristic polynomial of C is precisely the last invariant polynomial pr (λ ) of C. We decompose the invariant polynomials dd12(λλ),( ) ,, dr ( λ) of the λ-matrix A(λ ) into irreducible factors over the number field P kk11 12 k1s d1(λ) = pp 12( λλ) ( ) ps ( λ), d(λ) = ppkk21 ( λλ) 22 ( ) pk2s ( λ), 2 12 s kkrr12 krs drs(λ) = pp12( λλ) ( ) p( λ). Here, pp12(λλ),( ) ,, ps ( λ) are all the distinct irreducible polynomials over P (and with highest coefficient 1) that occur in dd12(λλ),( ) ,, dr ( λ) and k12j≤ k j ≤≤ kjrj , = 1, , s. All the power among kk11 12 krs pp12(λλ),( ) ,, ps ( λ) as far as they are distinct from 1, are called the elementary divisors of the λ-matrix A(λ ) in the number field P.
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