# linear algebra – Finding the JCF if we know the minimal polynomial and the charactristic polinomyal

Finde the Jordan canonical form of the real matrix $$A$$ with charecteristic polinomyal $$f(x)=a_{13}x^{13}+a_{12}x^{12}+ a_{11}x^{11}+ a_{10}x^{10} + a_{9}x^{9} + a_{8}x^{8} + a_{7}x^{7} + a_{6}x^{6}+a_{5}x^{5}+a_{4}x^{4}+ a_{3}x^{3}+a_{2}x^{2}+ a_{1}x + a_{0}$$ and minimal polinomyal $$(x-5)^{5}(x+8)^{4}(x-13)$$.

First, note that the grade of the minimal polynomal is $$10$$ and the charecteristic is $$3$$ so, we can have that this tres vectors correspond to one of the eigenvalues, or the three of them could be distributed in this three eigenvalues, then we can’t have one representation (excluiding the a rearrangement of the order of the Jordan blocks), we only can know the the sizes of the largest
Jordan blocks corresponding to each eigenvalue, Am I wrong?