Question

Orthogonally diagonalize the matrices in Exercises 13-22, giving an orthogonal matrix P and a diagonal matrix D. To save you time, the eigenvalues in Exercises 17-22 are the following:

Answer 1

To orthogonally diagonalize a matrix, find the eigenvalues and eigenvectors, arrange the eigenvectors in an orthogonal matrix, and the eigenvalues in a diagonal matrix. Use the equation A = PDP^T to decompose the original matrix.

Step-by-step explanation:

In order to orthogonally diagonalize a matrix, we need to find the eigenvalues and eigenvectors of the matrix. The eigenvectors will then form the columns of the orthogonal matrix, and the eigenvalues will be the diagonal elements of the diagonal matrix.

To find the eigenvalues, we set up the equation (A - λI)x = 0, where A is the matrix, λ is the eigenvalue, I is the identity matrix, and x is the eigenvector. Solving this equation will give us the eigenvalues. Once we have the eigenvalues, we can find the eigenvectors by solving the equation (A - λI)x = 0 for each eigenvalue.

Once we have the eigenvalues and eigenvectors, we can arrange the eigenvectors as columns in the orthogonal matrix P, and the eigenvalues in the diagonal matrix D. The original matrix can then be decomposed as A = PDP^T, where P^T is the transpose of P.