In linear algebra, the companion matrix of the monic polynomial Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

is the square matrix defined as For the square matrix section, see square matrix. ...

(While some authors use the transpose of this matrix, Wikipedia uses the above convention.) In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

The characteristic polynomial as well as the minimal polynomial of C(p) are equal to p; in this sense, the matrix C(p) is the "companion" of the polynomial p. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. ...

If the polynomial p(t) has n different zeros λ₁,...,λ_n (the eigenvalues of C(p)), then C(p) is diagonalizable as follows: In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

where V is the Vandermonde matrix corresponding to the λ's. In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with a geometric progression in each row, i. ...

If A is an n-by-n matrix with entries from some field K, then the following statements are equivalent: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

• A is similar to a companion matrix over K
• the characteristic polynomial of A coincides with the minimal polynomial of A
• there exists a vector v in Kⁿ such that {v, Av, A²v,...,A^n-1v} is a basis of Kⁿ

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 of A. Several equivalence relations in mathematics are called similarity. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In linear algebra, the Frobenius normal form of a matrix is a normal form that reflects the structure of the minimal polynomial of a matrix. ...