# Frobenius matrix

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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.

How to Cite This Entry:
Frobenius matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_matrix&oldid=12792
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article