\#4.) Let the matrix A below act on \( \mathrm{C}^{2} \). Find the complex-number eigenvalues and complex- number eigenvectors of matrix A . \[ A=\left[\begin{array}{cc}4 & 3 \\ -3 & 4\end{array}\right] \]

To find the eigenvalues and eigenvectors of the matrix \( A = \begin{pmatrix} 4 & 3 \\ -3 & 4 \end{pmatrix} \), we first compute the characteristic polynomial by finding the determinant of \( A - \lambda I \): \[ A - \lambda I = \begin{pmatrix} 4 - \lambda & 3 \\ -3 & 4 - \lambda \end{pmatrix} \] The determinant is calculated as: \[ \text{det}(A - \lambda I) = (4 - \lambda)(4 - \lambda) - (3)(-3) = (4 - \lambda)^2 + 9 \] \[ = \lambda^2 - 8\lambda + 16 + 9 = \lambda^2 - 8\lambda + 25 \] Next, we find the roots of the characteristic polynomial \( \lambda^2 - 8\lambda + 25 = 0 \) using the quadratic formula: \[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 - 100}}{2} = \frac{8 \pm \sqrt{-36}}{2} \] \[ = \frac{8 \pm 6i}{2} = 4 \pm 3i \] The eigenvalues are \( \lambda_1 = 4 + 3i \) and \( \lambda_2 = 4 - 3i \). To find the eigenvectors corresponding to each eigenvalue, we can substitute back into \( (A - \lambda I) \mathbf{v} = 0 \). For \( \lambda_1 = 4 + 3i \): \[ A - (4 + 3i)I = \begin{pmatrix} 3 - 3i & 3 \\ -3 & 3 - 3i \end{pmatrix} \] Setting up the equation \( \begin{pmatrix} 3 - 3i & 3 \\ -3 & 3 - 3i \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 0 \): We can use the first row: \[ (3 - 3i)x_1 + 3x_2 = 0 \Rightarrow x_2 = -\frac{(3 - 3i)}{3} x_1 = -(1 - i)x_1 \] Thus, the eigenvector for \( \lambda_1 \) can be expressed as: \[ \mathbf{v_1} = \begin{pmatrix} 1 \\ -1 + i \end{pmatrix} \] For \( \lambda_2 = 4 - 3i \): \[ A - (4 - 3i)I = \begin{pmatrix} 3 + 3i & 3 \\ -3 & 3 + 3i \end{pmatrix} \] Following the same process, we set up the equation: \[ (3 + 3i)x_1 + 3x_2 = 0 \Rightarrow x_2 = -\frac{(3 + 3i)}{3} x_1 = -(1 + i)x_1 \] Making the eigenvector: \[ \mathbf{v_2} = \begin{pmatrix} 1 \\ -1 - i \end{pmatrix} \] In summary, the eigenvalues of the matrix \( A \) are \( 4 + 3i \) and \( 4 - 3i \), with corresponding eigenvectors \( \begin{pmatrix} 1 \\ -1 + i \end{pmatrix} \) and \( \begin{pmatrix} 1 \\ -1 - i \end{pmatrix} \).