View source for Frobenius matrix

''companion matrix''

For every [[Polynomial|polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202001.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202002.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202003.png" /> such that the [[Characteristic polynomial|characteristic polynomial]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202005.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202006.png" />. Indeed, two such are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202008.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

These two matrices are similar and their minimal polynomial (cf. [[Minimal polynomial of a matrix|Minimal polynomial of a matrix]]) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202009.png" />, i.e. their similarity invariants are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020010.png" /> (see [[Normal form|Normal form]]). Both are called the companion matrix, or Frobenius matrix, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020011.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020012.png" /></td> </tr></table>

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., [[#References|[a1]]], p. 169.

For the matrix (a1), the first standard basis vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020013.png" /> is a [[Cyclic vector|cyclic vector]] (see also [[Pole assignment problem|Pole assignment problem]]). The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020014.png" /> form a so-called Krylov sequence of vectors for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020015.png" />, that is, a sequence of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020018.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020019.png" /> are independent, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020020.png" /> is a linear combination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020021.png" />.

The first and second natural canonical forms of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020022.png" /> (see also [[Normal form|Normal form]]) are block-diagonal with companion matrices as blocks. Both are also known as the Frobenius normal form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020023.png" />.

In a completely different setting, the phrase  "Frobenius matrix"  refers to a matrix giving the (induced) action of the [[Frobenius endomorphism|Frobenius endomorphism]] of an algebraic variety of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020024.png" /> on, say, the cohomology of that variety.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Stoer,   R. Bulirsch,   "Introduction to linear algebra" , Springer  (1993)  pp. Sect. 6.3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Marcus,   H. Minc,   "A survey of matrix theory and matrix inequalities" , Dover  (1992)  pp. Sect. I.3</TD></TR></table>