Exampile: Let A be a matrix \( \left(\begin{array}{ll}2 & 4 \\ 4 & 3\end{array}\right) \). Find an orthogonal basis of \( R^{2} \) O. consisting of eigenvector of \( A \)

To find an orthogonal basis of \( R^{2} \) consisting of eigenvectors of the matrix \( A = \left(\begin{array}{ll}2 & 4 \\ 4 & 3\end{array}\right) \), we first need to calculate its eigenvalues. The characteristic polynomial is obtained from the determinant of \( A - \lambda I \), where \( I \) is the identity matrix. Solving the equation gives us the eigenvalues \( \lambda_1 \) and \( \lambda_2 \). Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation \( (A - \lambda I)v = 0 \) for each eigenvalue \( \lambda \). After obtaining the eigenvectors, we can apply the Gram-Schmidt process to ensure they are orthogonal, yielding our desired orthogonal basis in \( R^{2} \). To ensure numerical accuracy, do not forget to check for arithmetic errors when calculating determinants and performing eigenvalue calculations, as these small mistakes can lead to incorrect eigenvalues and subsequently wrong eigenvectors. Always double-check your calculations, especially sign changes or matrix operations!