The companion matrix to a monic polynomial

(1)

is the square matrix

(2)

with ones on the subdiagonal and the last column given by the coefficients of . Note that in the literature, the companion matrix is sometimes
defined with the rows and columns switched, i.e., the transpose
of the above matrix.

When
is the standard basis , a companion matrix satisfies

(3)

for ,
as well as

(4)

including

(5)

The matrix minimal polynomial of the companion matrix is therefore , which is also its characteristic
polynomial .

Companion matrices are used to write a matrix in rational canonical form . In fact, any matrix whose matrix
minimal polynomial
has polynomial degree is similar to the companion
matrix for .
The rational canonical form is more interesting
when the degree of
is less than .

The following Wolfram Language command gives the companion matrix for a polynomial in the variable .

CompanionMatrix[p_, x_] := Module[
{n, w = CoefficientList[p, x]},
w = -w/Last[w];
n = Length[w] - 1;
SparseArray[{{i_, n} :> w[[i]], {i_, j_} /;
i == j + 1 -> 1}, {n, n}]]
See also Matrix ,

Matrix Minimal Polynomial ,

Rational Canonical
Form
This entry contributed by Todd
Rowland

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Cite this as:
Rowland, Todd . "Companion Matrix." From MathWorld --A Wolfram Web Resource, created by Eric
W. Weisstein . https://mathworld.wolfram.com/CompanionMatrix.html

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