Characteristic matrix, similarity invariants, minimum polynomial, companion matrix, non-derogatory matrix



Def. Characteristic matrix of a matrix. Let A be an nxn matrix whose elements are numbers from some number field F. The characteristic matrix of matrix A is the λ-matrix

λI - A.


Properties of the characteristic matrix λI - A of a matrix A. If A is an nxn matrix over a field F its characteristic matrix λ I - A has the following special properties:


● it is necessarily non-singular (i.e. it has a rank of n)

● it possesses n invariant factors (i.e. the diagonal of its Smith normal form is full – all diagonal elements contain monic polynomials (invariant factors) – there are no trailing zeros)



Elementary divisors of λI - D where D is a diagonal matrix. Let D be an

n-square diagonal matrix

 

                                        ole.gif



The elementary divisors of D’s characteristic matrix λI - D


                     ole1.gif


 are its diagonal elements λ - a1, λ - a2 , .... , λ - an.





Condition for similarity of two n-square matrices. Two n-square matrices A and B over a field F are similar over a field F if and only if their characteristic matrices have the same invariant factors or the same rank and the same elementary divisors in F.



Condition for similarity to a diagonal matrix. An n-square matrix A over a field F is similar to a diagonal matrix if and only if λ I - A has linear elementary divisors in F[λ].



Def. Similarity invariants of a matrix. Let A be an nxn matrix whose elements are numbers from some number field F. The similarity invariants of matrix A are the invariant factors of its characteristic matrix λI - A. Because the characteristic matrix λI - A of A is non-singular and has n invariant factors, matrix A has n similarity invariants.



Theorem. The characteristic polynomial of an n-square matrix A is the product of the invariant factors of λI - A (or, equivalently, of the similarity invariants of A).




The minimum polynomial and minimum equation of a matrix. Every square matrix satisfies its own characteristic equation (Hamilton-Cayley Theorem), however, there may be another equation of lower degree than the characteristic equation which the matrix satisfies. That monic polynomial m(λ) of minimum degree such that m(A) = 0 is called the minimum polynomial of A and m(A) = 0 is called the minimum equation of matrix A.


● Let λ1, λ2, .... , λl be the set of all distinct roots of the characteristic equation of a matrix A, and n1, n2, .... , nl be the multiplicities with which they occur. Then the characteristic polynomial, f(λ), of the matrix is


                    ole2.gif


The minimum polynomial, m(λ), is then:


                    ole3.gif


where 1 ≤mi ≤ni           i = 1,2,....,l



Theorems.


1] Let f(λ) be the characteristic polynomial and m(λ) be the minimum polynomial of an nxn matrix A. Then

                                       ole4.gif


where g(λ) is the greatest common divisor of all minors of order n-1 of the determinant |λI - A| .


2] When the characteristic roots of a matrix are all distinct the minimum polynomial is equal to (identical to) the characteristic matrix.


3] Let the characteristic matrix λ I - A of an nxn matrix A have the n monic polynomials p1(λ), p2(λ), .... , pn(λ) as the 1st, 2nd , ... , n th diagonal elements of its Smith normal form (i.e. these n monic factors represent its n invariant factors arranged in order). Then the minimum polynomial of A is pn(λ).


4] The characteristic matrix and minimum polynomial of an nxn matrix A are identical if and only if A has just one non-trivial similarity invariant.


5] If A is any n-square matrix over field F and f(λ) is any polynomial over F, then f(A) = 0 if and only if the minimum polynomial m(λ) of A divides f(λ).


6] The characteristic polynomial f(λ) of A is the product of the minimum polynomial of A and certain monic factors of m(λ).


7] The characteristic matrix of an n-square matrix A has distinct linear elementary divisors if and only if m(λ), the minimum polynomial of A, has only distinct linear factors.




Non-derogatory matrix. An nxn matrix whose characteristic polynomial and minimum polynomial are identical.


Derogatory matrix. An nxn matrix whose characteristic polynomial and minimum polynomial are different.



● An n-square matrix A is non-derogatory if and only if A has just one non-trivial similarity invariant.


● If matrices B1 and B2 have minimum polynomials m1(λ) and m2(λ) respectively, the minimum polynomial m(λ) of the direct sum D = diag(B1 ,B2 ) is the least common multiple of m1(λ) and m2(λ).


This result may be extended to the direct sum of n matrices.



● Let g1(λ), g2(λ), .... , gm(λ) be distinct, monic, irreducible polynomials in F[λ] and let Aj be a non-derogatory matrix such that 


   ole5.gif     


Then B = diag(A1, A2, .... , Am) has


    ole6.gif  


as both characteristic and minimum polynomial.





We will now exhibit a matrix which has as both its characteristic polynomial and minimum polynomial the polynomial


                            ole7.gif


Companion matrix of a monic polynomial. Let


                        ole8.gif


be a monic polynomial. We define the companion matrix of g(λ) as the nxn matrix C(g)


     C(g) = [-a1] if g(λ) = λ + a1 (i.e. if n = 1)



         ole9.gif         




This companion matrix C(g) of the polynomial g(λ) has g(λ) as both its characteristic polynomial and minimum polynomial. This means that if some non-derogatory matrix A has as its single non-trivial similarity invariant the polynomial g(λ) it will be similar to the companion matrix C(g) of g(λ) . Indeed C(g) of g(λ) will constitute a canonical form for such a matrix.



● If A is non-derogatory with the non-trivial similarity invariant fn(λ) = (λ - a)n, then

the matrix



     ole10.gif  




has fn(λ) as its characteristic and minimum polynomial.






________________________________________________________________________


Note. There are a variety of concepts associated with both regular matrices and

               Lambda-matrices:



            Regular n-square matrices (elements are numbers)


                  Characteristic matrix of a matrix A. The matrix λI - A .

                   

                  Similarity invariant of a matrix A. An invariant factor of λI - A .


                  Characteristic polynomial of a matrix A.. The polynomial f(λ) = | λI - A |

                                 (obtained by expanding the determinant | λI - A | ). It is a polynomial

                                  of degree n where A is nxn.. 


                  Characteristic equation of a matrix A. The equation f(λ) = | λI - A | = 0 .


                  Characteristic roots of a matrix A. The roots of the characteristic equation

                        f(λ) = | λI - A | = 0 . 

                  

                   Minimum polynomial and minimum equation of a matrix A. That monic

                                      polynomial m(λ) = 0 of minimum degree such that m(A) = 0.



            Lambda-matrices (elements are polynomials)

 

                    Invariant factors of a λ-matrix The monic polynomials f1(λ), f2(λ), .... , fr(λ)

                              in the diagonal of the Smith normal form of the matrix.


                    Elementary divisors of a λ-matrix. Invariant factors in Smith normal form

                                      expressed as irreducible factors.



________________________________________________________________________







References

  Ayres, Matrices. Chap. 25

  James and James. Mathematics Dictionary.



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