Chapter 7 - Eigenvalues and Eigenvectors - 7.6 Jordan Canonical Forms - Problems - Page 486: 7

55

A $10 \times 10$ matrix $A$ has $\lambda =2$ with multiplicity $4$, then there will be 5 possible Jordan block sizes: $ 4\\ 3,1\\ 2,2\\ 2,1,1\\ 1,1,1,1 $ Then $\lambda =5$ has multiplicity $6$, so there will be 11 possible Jordan block sizes: $6\\ 5,1\\ 4,2\\ 4,1,1\\ 3,3 \\ 3,2,1\\ 3,1,1,1 \\ 2,2,2 \\ 2,2,1,1\\ 2,1,1,1,1\\ 1,1,1,1,1,1$ Hence, the number of Jordan canonical forms is: $5 \times 1 =55$