Matrix Representation
The given matrix \( A \) is:
\[
A = \begin{pmatrix}
1 & 2 & 3 \\
0 & 2 & 3 \\
0 & 0 & 2
\end{pmatrix}
\]
This is an upper triangular matrix, which means the eigenvalues can be directly derived from its diagonal entries.
Characteristic Roots (Eigenvalues)
- The characteristic roots (or eigenvalues) of a matrix are found by solving the characteristic equation:
\[
\text{det}(A - \lambda I) = 0
\]
where \( I \) is the identity matrix and \( \lambda \) represents the eigenvalues.
- For the upper triangular matrix \( A \), the eigenvalues correspond to the diagonal elements.
Eigenvalue Calculation
- The diagonal elements of matrix \( A \) are:
- \( 1 \)
- \( 2 \)
- \( 2 \)
- Thus, the eigenvalues of the matrix are:
- \( \lambda_1 = 1 \)
- \( \lambda_2 = 2 \)
- \( \lambda_3 = 2 \)
Multiplicity of Eigenvalues
- The eigenvalue \( 2 \) has a multiplicity of \( 2 \) since it appears twice on the diagonal.
Conclusion
- The characteristic roots (eigenvalues) of the matrix \( A \) are:
- \( \lambda_1 = 1 \)
- \( \lambda_2 = 2 \) (multiplicity 2)
These eigenvalues indicate the scaling factors of the matrix transformations in their respective eigenspaces. Understanding these roots is essential for applications in stability analysis, system dynamics, and various fields of engineering and science.