Frobenius matrix

companion matrix

For every polynomial $f = \lambda ^ { n } + a _ { n - 1 } \lambda ^ { n - 1 } + \ldots + a _ { 1 } \lambda + a _ { 0 }$ there are $( n \times n )$-matrices $A$ such that the characteristic polynomial of $A$, $\operatorname { det } ( \lambda I - A )$, is equal to $f$. Indeed, two such are:

\begin{equation} \tag{a1} \left( \begin{array} { c c c c } { 0 } & { \square } & { \square } & { - a _ { 0 } } \\ { 1 } & { \ddots } & { \square } & { - a _ { 1 } } \\ { \square } & { \ddots } & { 0 } & { \vdots } \\ { \square } & { \square } & { 1 } & { - a _ { n - 1 } } \end{array} \right) \end{equation}

and

\begin{equation} \tag{a2} \left( \begin{array} { c c c c } { 0 } & { 1 } & { \square } & { \square } \\ { \square } & { \ddots } & { \ddots } & { \square } \\ { \square } & { \square } & { 0 } & { 1 } \\ { - a _ { 0 } } & { \cdots } & { \cdots } & { - a _ { n - 1 } } \end{array} \right). \end{equation}

These two matrices are similar and their minimal polynomial (cf. Minimal polynomial of a matrix) is $f$, i.e. their similarity invariants are $1 , \dots , f$ (see Normal form). Both are called the companion matrix, or Frobenius matrix, of $f$.

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

\begin{equation*} \left( \begin{array} { c c c } { A _ { 1 } } & { \square } & { * } \\ { \square } & { \ddots } & { \square } \\ { 0 } & { \square } & { A _ { n } } \end{array} \right) \end{equation*}

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 $e_1$ is a cyclic vector (see also Pole assignment problem). The vectors $e _ { 1 } , \dots , e _ { n } , - ( a _ { 0 } e _ { 1 } + \ldots + a _ { n - 1} e _ { n } )$ form a so-called Krylov sequence of vectors for $A$, that is, a sequence of vectors $v _ { 1 } , \dots , v _ { n + 1 }$ such that $A v _ { i } = v _ { i + 1}$, $i = 1 , \dots , n$, the $v _ { 1 } , \dots , v _ { n }$ are independent, and $v _ { n+1 } = A v _ { n}$ is a linear combination of $v _ { 1 } , \dots , v _ { n }$.

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

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 $p > 0$ 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