We give a proof by induction that any complex-valued matrix A may be put in Jordan normal form. The 1 × 1 case is trivial. Let "A" be an "n" × "n" matrix. Take any eigenvalue λ of "A". The range of "A" − λ I, denoted by Ran("A" − λ I), is an invariant subspace of "A". Also, since λ is an eigenvalue of "A", the dimension of Ran("A" − λ I), "r", is strictly less than "n". Let "A' " denote the restriction of "A" to Ran("A" − λ I), by inductive hypothesis, there exists a basis {"p"1, …, "p""r"} such that "A' ", expressed with respect to this basis, is in Jordan normal form.