The matrix
J=
\begin{pmatrix}
A11 & \ast & A13 \\
A21 & A22 & \otimes \\
A31 & A32 & A33
\end{pmatrix}
is the Jordan normal form of a matrix A, that is not diagonalizable.
Determine the entries \ast und \otimes.
\ast
=
0
1
\otimes
=
1
0
Since the matrix A is not diagonalizable, it cannot be
\ast= 0 = \otimes the case.
The diagonal entries L1 and (double) L2 are the eigenvalues
of A.
The diagonal entries L2 and (double) L1 are the eigenvalues
of A.
Due to the non-diagonalizability, the double eigenvalue L2 has a Jordan block of length 2.
Due to the non-diagonalizability, the double eigenvalue L1 has a Jordan block of length 2.
Therefore, it follows that \ast = 0 and \otimes = 1.
Therefore, it follows that \ast = 1 and \otimes = 0.