Chapter 7 - Eigenvalues and Eigenvectors - 7.7 Chapter Review - Additional Problems - Page 490: 15

See below

There are 10 possible Jordan canonical forms: $J_1=\begin{bmatrix} 6 & 0 & 0 & 0 & 0 & 0\\ 0 & 6 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & 6 & 0\\ 0 & 0 & 0 & 0 & 0 & 6\\ \end{bmatrix}$ There are 6 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $1$ $J_2=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 5 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $2$ $J_3=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 1 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 4 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $2$ $J_4=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 1 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 3 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $2$ $J_5=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 1 & 0 & 0 & 0\\ 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 4 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $3$ $J_6=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 1 & 0 & 0 & 0\\ 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 1\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 3 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $3$ $J_7=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 1 & 0 & 0 & 0\\ 0 & 0 & 6 & 1 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 3 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $4$ $J_8=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 1 & 0 & 0 & 0\\ 0 & 0 & 6 & 1 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 1\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 2 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $4$ $J_9=\begin{bmatrix} 6 & 1 & 0 & 0 & 0 & 0\\ 0 & 6 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 1\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 4 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $2$ $J_2=\begin{bmatrix} 6 & 0 & 0 & 0 & 0 & 0\\ 0 & 6 & 0 & 0 & 0 & 0\\ 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 6 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 1\\ 0 & 0 & 0 & 0 & 0 & -3\\ \end{bmatrix}$ There are 5 linearly independent eigenvectors of a matrix with this Jordan canonical form, and the maximum length of a cycle is $2$