Let "V" be a vector space over a field "K". Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in "K", or equivalently if the characteristic polynomial of the operator splits into linear factors over "K". This condition is always satisfied if "K" is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.