# Jordan form, similarity and eigenvalues

A complex matrix is always diagonalizable in the Jordan canonical form (right? ). also, two matrices have the same Jordan form if and only if they are similar. But two matrices who have the same eigenvalues might not be similar. SO, if A and B have the same eigenvalues, I say that they have the same Jordan blocks, so the same Jordan Form. But I am wrong, cause that would mean that they are similar. So could anyone explain me where I am wrong?

• We don't say diagonalizable in JNF. We say similar to a JNF or something of the sort. The mistake is saying that they have the same jordan blocks. – Git Gud Mar 29 '15 at 18:08
• could you explain a bit more please? why don t they have the same jordan blocks? cause a block is formed with eigenvalues . so , where am i wrong? – Dan Leonte Mar 29 '15 at 18:08
• The same eigenvalue can have several jordan blocks. – Git Gud Mar 29 '15 at 18:26

Your main error is to confuse diagonalisability and having a Jordan normal form. Not all complex matrices are diagonalisable. Most Jordan normal forms are not diagonal matrices, and those are not diagonalisable either. Jordan block can have different eigenvalues and different sizes. Only if all Jordan blocks have size$~1$ is the matrix diagonalisable.
In terms of classifying matrices up to similarity, there is much more than just the set of eigenvalues. In order to be similar two matrices must have identical characteristic polynomials, so the same eigenvalues and the same algebraic multiplicities for each eigenvalue. But this is not sufficient: for each eigenvalue$~\lambda$ one can have a collection of Jordan blocks for$~\lambda$ whose sizes add up to the algebraic multiplicity of$~\lambda$. There is no imposed order of this collection of blocks, so one is free to order them say by weakly decreasing size so as to obtained a partition of the algebraic multiplicity; only if for each eigenvalue one obtained the same partition of the (same) algebraic multiplicity can one conclude that matrices are similar.