case is a zero matrix. The solution with minimum Euclidean norm is This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let . , we have where and denotes the Frobenius norm. Obtaining all solutions of a linear system If the linear system has any solutions, they are all given by for arbitrary vector . Solution(s) exist if and only if . If the latter holds, then the solution is unique if and only if has full column rank, in which case is a zero matrix. If solutions exist but does not have full column rank, then we have an indeterminate system, all of whose infinitude of solutions are given by this last equation. Minimum norm solution to a linear system For linear systems with non-unique solutions (such as under-determined systems), the pseudoinverse may be used to construct the solution of minimum Euclidean norm among all solutions. If is satisfiable, the vector is a solution, and satisfies for all solutions. This result is easily extended to systems with multiple right-hand sides, when the Euclidean norm is replaced by the Frobenius norm. Let . If is satisfiable, the matrix is a solution, and satisfies for all solutions. Condition number Using the pseudoinverse and a matrix norm, one can define a condition number for any matrix: A large condition number implies that the problem of finding least-squares solutions to the corresponding system of linear equations is ill-conditioned in the sense that small errors in the entries of can lead to huge errors in the entries of the solution. Generalizations Besides for matrices over real and complex numbers, the conditions hold for matrices over biquaternions, also called "complex quaternions". In order to solve more general least-squares problems, one can define Moore–Penrose inverses for all continuous linear operators between two Hilbert spaces and , using the same four conditions as in our definition above. It turns out that not every continuous linear operator has a continuous linear pseudoinverse in this sense. Those that do are precisely the ones whose range is closed in . A notion of pseudoinverse exists for matrices over an arbitrary field equipped with an arbitrary involutive automorphism. In this more general setting, a given matrix doesn't always have a pseudoinverse. The necessary and sufficient condition for a pseudoinverse to exist is that , where denotes the result of