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Eigenvalues and eigenvectors play a crucial role in understanding matrices. An eigenvalue is a number that describes the factor by which a linear transformation alters a vector. The corresponding eigenvector is the vector that undergoes this transformation while remaining in the same direction.
Consider the matrix transformation represented by matrix \( S \). In linear algebra, if \( A \) is our transformation matrix, the equation \( Av = lambda v \) holds, where \(v\) is an eigenvector and \( lambda \) is the corresponding eigenvalue.
In the context of the exercise, matrix \( S \) is expressed similar to matrix \( D \), implying it preserves eigenvalues this way. \( D \), being diagonal, has straightforward eigenvalues, which are just the entries on its diagonal: 2 and 5. Therefore, these are also eigenvalues of \( S \).
Finding eigenvectors involves solving \( (S - lambda I)v = 0 \), where \( I \) is the identity matrix. For diagonal matrices, eigenvectors correspond to each axis: \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \) and \( \begin{bmatrix} 0 \ 1 \end{bmatrix} \) correspond to eigenvalues 2 and 5. Since \( S \) is constructed by rotating \( D \), the eigenvectors of \( S \) are rotated versions of these standard basis vectors. Thus, they are \( \begin{bmatrix} \cos \theta \ \sin \theta \end{bmatrix} \) and \( \begin{bmatrix} -\sin \theta \ \cos \theta \end{bmatrix} \).

Matrix theory provides a fundamental understanding of linear transformations in math. A matrix is a rectangular arrangement of numbers or functions used for transformations. Through manipulation, matrices can illustrate systems of equations simultaneously.

• Rotation Matrix: One interesting type of matrix is the rotation matrix. It rotates points in a plane. Rotation matrices are orthogonal, meaning their inverse is their transpose.
• Diagonal Matrix: Another key player in matrix theory is the diagonal matrix. It simplifies operations as its non-diagonal entries are zero. The ease of eigenvalue computation is due to the direct representation of values on the diagonal.

The given problem utilizes a rotation matrix and a diagonal matrix. By expressing \( S \) as \( M_1 \, D \, M_2 \), where both \( M_1 \) and \( M_2 \) are rotation matrices and \( D \) is diagonal, we simplify our understanding of the problem.
The manipulation stems from realizing the multiplication of these structured matrices doesn't alter the rotation, allowing us to derive properties without matrix multiplication.

A matrix is symmetric if it equals its transpose, meaning \( A = A^T \). Symmetric matrices have nicer properties, especially in computations and applications like optimization and physics.
A matrix \( A \) is also considered positive definite if all its eigenvalues are positive, which generally implies specific beneficial properties like ensuring unique solutions and stability in systems.

• **Symmetric Nature of S:** In our problem, matrix \( S \) maintains symmetry due to the construction with symmetric matrices, ensuring the eigenvectors remain orthogonal.
• **Positive Definiteness:** It means matrix \( S \) will map any non-zero vector \( x \) to a positive dot product \( x^T S x > 0 \).

Since the eigenvalues of \( S \) are 2 and 5, both positive, plus the symmetry derived from the rotation matrix and diagonal matrix combination, \( S \) is confirmed to be symmetric positive definite.