Word Count : 807

---

To find the eigenvalues and orthonormal eigenvectors of the given real symmetric matrix: \[ A = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ \sin\theta & -\cos\theta & 0 \\ 0 & 0 & -1 \end{bmatrix}, \] we proceed step by step. --- ### Step 1: Find the eigenvalues The eigenvalues \(\lambda\) are found by solving the characteristic equation: \[ \det(A - \lambda I) = 0, \] where \(I\) is the identity matrix. Substituting \(A\) and \(I\), we get: \[ A - \lambda I = \begin{bmatrix} \cos\theta - \lambda & \sin\theta & 0 \\ \sin\theta & -\cos\theta - \lambda & 0 \\ 0 & 0 & -1 - \lambda \end{bmatrix}. \] The determinant of this matrix is: \[ \det(A - \lambda I) = \begin{vmatrix} \cos\theta - \lambda & \sin\theta & 0 \\ \sin\theta & -\cos\theta - \lambda & 0 \\ 0 & 0 & -1 - \lambda \end{vmatrix}. \] Expanding the determinant along the third row, we get: \[ \det(A - \lambda I) = (-1 - \lambda) \cdot \begin{vmatrix} \cos\theta - \lambda & \sin\theta \\ \sin\theta & -\cos\theta - \lambda \end{vmatrix}. \] The \(2 \times 2\) determinant is: \[ \begin{vmatrix} \cos\theta - \lambda & \sin\theta \\ \sin\theta & -\cos\theta - \lambda \end{vmatrix} _____ _______ __________ _______ ______ _________ ________ ___.
______ ___ _______ ______ ___ ___ ___ _____ ______ ___ __________.
___ _______ ________ ___ ______ _____ ___ ______ ____ ___ ____ _________.
____ _____ ______ ________ _______ ______ ______ _____.
_____ ______ ____ ____ ___ ____ ______ ___ _______ _________ __________ _______.
_______ _________ __________ _______ ___ ________.
____ ______ ______ ___ ________ ______ ____ _______ _______.
_____ _______ __________ ________ ______ _____ _________ _______.
_____ ______ ________ _____ __________ _______ _____ __________.
___ _______ ______ _____ _____ ____ ___.
___ __________ ______ ______ ______ ________ ___ __________ _____ ___.
_________ ___ _____ ____ ____ __________ ___ ___ _________.
_______ _____ ______ _______ ______ _____ _______.
______ _________ ____ _______ ___ _______ ____ _______ ________ _________ ____ _______.
_______ ___ ______ _______ ____ __________ ______ _______ _______.
_______ ________ _________ _________ ____.
___ __________ _____ ________ ________ ______ _________ _________ ________ ______ _____ ________.
____ __________ ___ _____ _________ _________ _______.
_________ _________ ________ _____ ___ __________ _________ _______ __________.
______ ____ ____ _______ __________ _________ ________ _________ _______ _________ ___ _____.
_________ _________ __________ _____ _______.
______ ___ __________ _____ ______ __________ __________ ____ _______ _________ _________ _________.
________ __________ __________ ____ __________ ___ ________ __________ _______ ______ _____ ________.
___ _______ ____ ______ ____ ________ __________.
_____ _________ _______ _______ ___ _____ __________ __________.
___ _________ _______ ________ _______ ___ __________ __________ ________ ______ _____.
_______ _______ __________ ______ _____ ____.
_________ _____ ______ ___ _______ _________ ____.
____ ________ _____ __________ __________ _______ __________ ____ _______.
_______ _________ _____ ______ _________ ______ _____ ________ ________.
________ _________ ______ _______ _______ ____.
____ _____ ________ ___ ______ ___ ___ ____ __________ ________ ____.
__________ ______ ___ ______ ________ ________ ___ ______ ________ _______ _______.
_________ __________ __________ ___ ________ ________ ________ _______ _________ ___ _______ _____.
____ _________ ______ ___ ____ ________ ________ _______ ___ _________ _________ ________.
_______ ________ ________ ________ _________ _________ ___ _______ ________ ___ _________.
_____ _______ ________ _____ ______ _____.
________ _________ __________ ___ __________ _______ __________ ________ ________.
____ ____ __________ _____ _______ ____ ____ ___ ____.
_________ _______ _________ _____ _____.
_____ ____ _________ _________ _________ ____ ______ ___ ____ ______.
_________ ___ __________ ___ ______ __________ ________ _________ _______ ________.
________ _________ _____ ___ ______ ________ ___ _______ _____ _____ ________ _________.
______ ______ _____ ________ __________ _____ ____.
_____ _____ _______ ________ __________ _________.
_________ ________ __________ ___ ___ ___ _______.
_____ ______ ______ __________ _______ _____ __________ _______.
___ ___ __________ ________ __________ ________ ________ ______ ____ ______ ________.
_____ ___ ____ ________ ________ _________ ___.
_________ _________ ________ _________ _____ ________ ______ _____.
_______ __________ _____ ____ ________ ___ ____ ________ ____ _____ __________ _______.
_______ ___ ______ ________ _______ ___ ____ ____ _____ _________.
______ _____ ________ _____ ________ ______ __________ ________ __________.
____ __________ ______ ____ _____ __________ ______ ________.
_________ _____ _________ __________ ____.
________ ____ ____ ______ ____ ________ _____ _______ ______ _____ ____ __________.
______ ______ ____ ___ ____ _______ ________.
________ ____ ____ ________ _____ ________.
________ _______ __________ __________ ________ __________ ___ _______ ____.
_______ ___ _________ __________ _________ _________ _______ _________ _____.
_______ _____ _______ ______ ________ ____ _______ ____ _____ _______ __________.
_____ _______ _________ _____ __________ __________ _______ ___ _________ ________ ____.
_________ __________ _________ _________ ____ ____ ________ ___ _______.
_____ ___ __________ _______ _____ ____ __________ _____ _______ ________ ___.
___ ____ _______ __________ ____ __________ ________ ____ _______ _________ ______ ____.
___ __________ ________ _______ _________ ______ _______.
___ _______ _________ ___ _______.
_____ ________ ________ _____ ________ ____ ___ ______.
Get Full Answer on WhatsApp