Final answer:
To diagonalize S = [9 12; 12 16], we need to find the eigenvalues and eigenvectors. The orthogonal matrices that diagonalize S are given by P^TSP, where P is the matrix formed by the normalized eigenvectors. The eigenvalues of S are 0 and 25.
Step-by-step explanation:
An orthogonal matrix is a square matrix whose transpose is equal to its inverse. In order to diagonalize the given matrix S = [9 12; 12 16], we need to find the eigenvalues and eigenvectors.
Step 1: Find the eigenvalues.
To find the eigenvalues, we solve the characteristic equation det(S - λI) = 0, where I is the identity matrix. Substituting the values of S, we get det([[9-λ, 12], [12, 16-λ]] = 0. Expanding the determinant and solving the equation, we find two distinct eigenvalues λ1 = 0 and λ2 = 25.
Step 2: Find the eigenvectors.
To find the corresponding eigenvectors, we substitute each eigenvalue into the equation (S - λI)x = 0 and solve for x. For λ1 = 0, we get the equation [[9, 12], [12, 16]]x1 = 0. Solving the system of linear equations, we find x1 = [-4/3, 1]. Similarly, for λ2 = 25, we get the equation [[-16, 12], [12, -9]]x2 = 0 and solve for x2, which results in x2 = [3/4, 1].
Step 3: Construct the orthogonal matrix.
The orthogonal matrix P is formed by taking the normalized eigenvectors as columns. P = [[-4/5, 3/5], [3/5, 4/5]].
Thus, the orthogonal matrices that diagonalize S = [9 12; 12 16] are given by PTSP, where P = [[-4/5, 3/5], [3/5, 4/5]].