Frobenius matrix

companion matrix

For every polynomial there are -matrices such that the characteristic polynomial of , , is equal to . Indeed, two such are:

(a1)

and

(a2)

These two matrices are similar and their minimal polynomial (cf. Minimal polynomial of a matrix) is , i.e. their similarity invariants are (see Normal form). Both are called the companion matrix, or Frobenius matrix, of .

More generally, a matrix of block-triangular form with as diagonal blocks one of the companion matrices above (all of the same type),

is also sometimes called a Frobenius matrix.

Somewhat related, a matrix with just one column (or one row, but not both) different from the identity matrix is also sometimes called a Frobenius matrix; see, e.g., [a1], p. 169.

For the matrix (a1), the first standard basis vector is a cyclic vector (see also Pole assignment problem). The vectors form a so-called Krylov sequence of vectors for , that is, a sequence of vectors such that , , the are independent, and is a linear combination of .

The first and second natural canonical forms of a matrix (see also Normal form) are block-diagonal with companion matrices as blocks. Both are also known as the Frobenius normal form of .

In a completely different setting, the phrase "Frobenius matrix" refers to a matrix giving the (induced) action of the Frobenius endomorphism of an algebraic variety of characteristic on, say, the cohomology of that variety.

References

[a1]	J. Stoer, R. Bulirsch, "Introduction to linear algebra" , Springer (1993) pp. Sect. 6.3
[a2]	M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities" , Dover (1992) pp. Sect. I.3