Matrices 3 x 3 over Z5
Task number: 2659
Determine eigenvalues and the corresponding eigenvectors for the following matrix over the field \(\mathbb Z_5\). Decide whether this matrix is diagonalizable:
\( \begin{pmatrix} 1 & 0 & 0 \\ 2 & 3 & 3 \\ 1 & 1 & 0 \\ \end{pmatrix} \)
Result
\(\lambda_1=2\), \(\mathbf x_{1}=c\cdot(0, 1, 3)^T\); \(\lambda_2=\lambda_3 = 1\), \(\mathbf x_2=c\cdot(1, 0, 1)^T\), \(\mathbf x_3=c\cdot(1, 4, 0)^T\);
It is diagonalizable. The diagonal form is e.g. \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix} \)
