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Canonical forms under similarity; Rational, Jacobson and Jordan canonical forms; hypercompanion matrix

Rational Canonical Form. Given an nxn matrix A, let Ci, Ci+1, .... , Cn be the companion matrices of the non-trivial invariant factors of λI - A. Then the rational canonical form for all matrices similar to A is

In other words, the rational form is the direct sum of the companion matrices Ci, Ci+1, .... , Cn:

S = diag (Ci, Ci+1, .... , Cn)

● Every square matrix A is similar to the direct sum of the companion matrices of the non-trivial invariant factors of λI - A.

A second canonical form. Given an nxn matrix A, let Ci, Ci+1, .... , Cn be the companion matrices of the elementary divisors of λI - A. Then a canonical form for all matrices similar to A is

In other words, the form is the direct sum of the companion matrices Ci, Ci+1, .... , Cn:

● Every square matrix A over F is similar to the direct sum of the companion matrices of the elementary divisors over F of λI - A.

Hypercompanion matrix. Let {p(λ)}q be one of the elementary divisors of the characteristic matrix of some λ-matrix and let C(p) be the companion matrix of p(λ). The hypercompanion matrix H associated with the elementary divisor {p(λ)}q is given by

where M is a matrix of the same order as C(p) having the element 1 in the lower left-hand corner and zeros elsewhere. The diagonal of the hypercompanion matrix H consists of q identical C(p) matrices. Note that there is a continuous line of 1's just above the diagonal.

● Every square matrix A over F is similar to the direct sum of the hypercompanion matrices of the elementary divisors over F of λI - A.

The Jacobson canonical form. The Jacobson canonical form of a square matrix A consists of the direct sum of the hypercompanion matrices of the elementary divisors over F of

λI - A i.e. the matrix J

where Hi is the hypercompanion matrix associated with the i-th elementary divisor.

The Classical (or Jordan) canonical form. Let the elementary divisors of the characteristic matrix of a matrix A be powers of linear polynomials. The canonical form is then the direct sum of hypercompanion matrices of the form

corresponding to the elementary divisor {p(λ)}q = ( λ - ai)q . The diagonal contains q identical ai’s. This special case of the Jabobson canonical form is known as the Jordan or Classical canonical form.

● Let F be the field in which the characteristic polynomial of a matrix A factors into linear polynomials. Then A is similar over F to the direct sum of hypercompanion matrices of the form (1) above, each matrix corresponding to an elementary divisor ( λ - ai)q .

● An n-square matrix A is similar to a diagonal matrix if and only if the elementary divisors of λI - A are linear polynomials, that is, if and only if the minimum polynomial of A is the product of distinct linear polynomials.

References

Ayres. Matrices.