Every square matrix is similar to an Jordan matrix, that is, for any square matrix A of order N, there is a reversible square matrix X of order N and Jordan matrix J of order N, so that A = X (-1) JX; If the matrix is a quasi-diagonal matrix composed of Jordan blocks, and if the blocks are matrices with the same elements on the main diagonal, all the elements on the diagonal above the main diagonal are 1, and all the other elements are 0. If a Jordan block can be decomposed into a quantity matrix+a nilpotent matrix, then the Jordan matrix can be decomposed into a diagonalizable matrix+a nilpotent matrix (the power here should be the common multiple of the powers of all nilpotent matrices decomposed by Jordan blocks contained in the Jordan matrix).

After decomposition, use X and X (- 1) to get the decomposition formula of A. ..

Uniqueness is because the Jordan canonical form of any matrix is unique regardless of the arrangement order of Jordan blocks, and the arrangement order is also fixed after x and x are multiplied (-1).

I don't think you understand what I mean. Give an example to show that the problem is the simplest. If you understand that it is standard, you can qq me, 22949520. I use an example to illustrate how I did it.