This can be used to show the uniqueness of the Jordan form. Let "J"1 and "J"2 be two Jordan normal forms of "A". Then "J"1 and "J"2 are similar and have the same spectrum, including algebraic multiplicities of the eigenvalues. The procedure outlined in the previous paragraph can be used to determine the structure of these matrices. Since the rank of a matrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of "J"1 and "J"2. This proves the uniqueness part of the statement.