Final answer:
The eigenvectors of an upper triangular matrix are determined by solving the system (A - λI)x = 0 for each eigenvalue λ. The Jordan normal form is a diagonal matrix if all eigenvalues are distinct, with Jordan blocks containing ones on the superdiagonal for repeated eigenvalues.
Step-by-step explanation:
The eigenvectors and Jordan normal form of an upper triangular matrix can be determined by examining the diagonal entries of the matrix, which represent the eigenvalues. Since an upper triangular matrix has its eigenvalues along the diagonal, they can be written directly as the entries of a diagonal matrix, which is a special case of the Jordan normal form. To find the eigenvectors for each eigenvalue, we solve the system of equations (A - λI)x = 0, where A is the matrix, λ is an eigenvalue, and I is the identity matrix. Usually, for a distinct eigenvalue of an upper triangular matrix, the eigenvector corresponding to it can be found by back substitution starting from the bottom row of the system.
Finding the Eigenvectors:
- For each eigenvalue λ (diagonal entries of A), construct the matrix A - λI.
- Solve the system (A - λI)x = 0 for each λ to get the corresponding eigenvectors.
The Jordan normal form of an upper triangular matrix is a diagonal matrix if all the eigenvalues are distinct. If there are repeated eigenvalues, the Jordan blocks corresponding to those eigenvalues will contain ones on the superdiagonal.