Endpmatrix

Endpmatrix is a term sometimes used to refer to the matrix representation of an endomorphism in linear algebra. An endomorphism is a linear map from a finite-dimensional vector space to itself. The endpmatrix encodes the action of that map in a chosen basis, allowing linear transformations to be treated as matrices acting on coordinate vectors.

To construct an endpmatrix, fix a basis for the vector space V of dimension n. If T: V → V is an endomorphism, its endpmatrix A is the n×n matrix whose j-th column consists of the coordinate vector of T(v_j) in that basis, where {v_1, ..., v_n} is the basis. The matrix A represents the transformation T with respect to that basis, and applying T to a vector v corresponds to multiplying the coordinate column [v] by A.

Changing the basis changes the endpmatrix by a similarity transformation: if P is the invertible matrix that changes from the old basis to the new basis, then the new endpmatrix A′ satisfies A′ = P^{-1}AP. Consequently, endpmatrices corresponding to the same endomorphism are similar, and they describe the same linear map in different coordinates.

The set of all endomorphisms End(V) forms a ring under addition and composition, and, once a basis is fixed, End(V) is isomorphic to the matrix ring M_n(F). Common examples include the zero endomorphism (endpmatrix 0), the identity endomorphism (endpmatrix I_n), projection maps, and nilpotent endomorphisms (which have nilpotent endpmatrices, such as Jordan blocks).

Note that the term endpmatrix is not universally standard; many texts simply refer to the matrix of an endomorphism or the matrix representation with respect to a basis. The concept, however, remains the same: it is the matrix form that captures how an endomorphism acts on coordinates.