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A linear map from a vector space to itself.
A linear operator is a linear map on a vector space .
Because the domain and codomain agree, one can form iterates using composition. Linear operators are the objects to which eigenvalues, eigenspaces, and the characteristic polynomial are attached.
A linear map is a function between vector spaces over the same field such that for all and ,
A vector space over a field is a set equipped with two operations (functions) and such that for all and :
Vector spaces are the basic objects studied via linear maps and invariants such as the determinant and eigenvalues of operators.
A composition of functions is the function obtained by applying one function after another: if and are functions (so the codomain of matches the domain of ), then the composition is defined by
An eigenvalue of a linear operator is a scalar such that there exists a nonzero vector with
A vector satisfying this equation is an eigenvector, and all such vectors (together with ) form the eigenspace for . Eigenvalues are precisely the roots of the characteristic polynomial.
An eigenspace of a linear operator associated to a scalar is the set
A characteristic polynomial of a linear operator on an -dimensional vector space is the polynomial
where is the matrix of in any basis and is the identity matrix. This definition is independent of the chosen basis.