A companion matrix in its standard form for a monic polynomial
 |
(1)
|
of polynomial degree 
is the 
square matrix
![A=[0 0 ... 0 -a_0; 1 0 ... 0 -a_1; 0 1 ... 0 -a_2; | | ... ... |; 0 0 ... 1 -a_(n-1)]](/images/equations/CompanionMatrix/NumberedEquation2.svg) |
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with 1s on the subdiagonal and the last column given by the negatives of 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.
In this convention, 
is the multiplication matrix of 
on the quotient ring ![K[x]/<a(x)>](/images/equations/CompanionMatrix/Inline6.svg)
in the ordered vector
basis 
,
, ..., 
, where 
is a field containing the coefficients
of 
.
When 
,
..., 
form the standard basis, a companion matrix satisfies
 |
(3)
|
for 
,
as well as
 |
(4)
|
including
 |
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The matrix minimal polynomial of the companion matrix is therefore 
. In the monic convention, its characteristic
polynomial is 
,
where 
is the identity matrix. The alternative convention
used by CharacteristicPolynomial[m,
x] gives 
.
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 companion matrix for a polynomial 
in the variable 
is implemented in the Wolfram
Language as CompanionMatrix[p,
x]. For example,
CompanionMatrix[x^3 + 2 x + 3, x]
returns
![[0 0 -3; 1 0 -2; 0 1 0].](/images/equations/CompanionMatrix/NumberedEquation6.svg) |
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|
The term companion matrix is also used more broadly for a matrix with a prescribed characteristic polynomial.
For a real-rooted polynomial 
of polynomial degree 
, Schmeisser (1993, pp. 13-14) constructs
a real symmetric matrix in tridiagonal
matrix form without first finding the roots. For distinct
roots, start with 
and 
, with 
a monic polynomial, and
perform successive polynomial divisions in
the form
 |
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where 
is a monic polynomial and 
for 
, ..., 
. Write 
and 
. Then
![T=[d_1 sqrt(c_1) 0 ...; sqrt(c_1) d_2 sqrt(c_2) ...; 0 sqrt(c_2) d_3 ...; | | | ...],](/images/equations/CompanionMatrix/NumberedEquation8.svg) |
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with
 |
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|
The construction extends to repeated roots, with zero off-diagonal entries separating smaller blocks.
For example, 
gives
![T=[0 sqrt(2/3) 0; sqrt(2/3) 0 1/sqrt(3); 0 1/sqrt(3) 0],](/images/equations/CompanionMatrix/NumberedEquation10.svg) |
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|
whose eigenvalues are 
, 0, and 1. With distinct roots,
is similar
to the standard companion matrix. This need not hold for repeated roots.
For 
,
the zero 
matrix has the required characteristic
polynomial, but its matrix minimal polynomial
is 
rather than 
.
It is therefore not similar to the standard companion
matrix.
