In linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, the companion matrix of the monic polynomial

is the square matrix defined as


With this convention, and writing the basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

 as , one has (for ), and generates V as a -module: C cycles basis vectors.

Some authors use the transpose

Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

 of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear recursive relations.

Characterization

The characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 as well as the minimal polynomial

Minimal polynomial (linear algebra)

In linear algebra, the minimal polynomial of an n-by-n matrix A over a field F is the monic polynomial P over F of least degree such that P=0...

 of C(p) are equal to p;
in this sense, the matrix C(p) is the "companion" of the polynomial p.

If A is an n-by-n matrix with entries from some field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 K, then the following statements are equivalent:

• A is similar to the companion matrix over K of its characteristic polynomial
• the characteristic polynomial of A coincides with the minimal polynomial of A, equivalently the minimal polynomial has degree n
• there exists a cyclic vector v in for A, meaning that {v, Av, A²v,...,A^n-1v} is a basis
  Basis (linear algebra)
  In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
  
   of V. Equivalently, such that V is cyclic as a -module (and ); one says that A is regular.

Not every square matrix is similar to a companion matrix. But every matrix is similar to a matrix made up of blocks of companion matrices. Furthermore, these companion matrices can be chosen so that their polynomials divide each other; then they are uniquely determined by A. This is the rational canonical form

Frobenius normal form

In linear algebra, the Frobenius normal form, Turner binormal projective form or rational canonical form of a square matrix A is a canonical form for matrices that reflects the structure of the minimal polynomial of A and provides a means of detecting whether another matrix B is similar to A...

 of A.

Diagonalizability

If p(t) has distinct roots λ₁,...,λ_n (the eigenvalues of C(p)), then C(p) is diagonalizable as follows:
where V is the Vandermonde matrix corresponding to the λ's.

Linear recursive sequences

Given a linear recursive sequence with characteristic polynomial
the (transpose) companion matrix
generates the sequence, in the sense that
It increments the series by 1.