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

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

 are its diagonal elements .

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 be the set of all distinct roots of the characteristic equation of a matrix A, and be the multiplicities with which they occur. Then the characteristic polynomial, , of the matrix is

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

where 1 ≤m_i ≤n_i           i = 1,2,....,l

Theorems.

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

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 as the 1 st, 2 nd , ... , 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 ,

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 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 and respectively, the minimum polynomial of the direct sum D = diag(B₁ ,B₂ ) is the least common multiple of and .

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

● Let be distinct, monic, irreducible polynomials in F[λ] and let be a non-derogatory matrix such that 

Then has

as both characteristic and minimum polynomial.

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

Companion matrix of a monic polynomial. Let

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

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

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 f_n(λ) = (λ - a)ⁿ, then

the matrix

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

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Note. We speak of a variety of things 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 = | 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 = | λI - A | = 0 .

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

                          = | 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

                              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.

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References

  Ayres, Matrices. Chap. 25

  James and James. Mathematics Dictionary.