Question

Let A be the matrix for reflecting through the plane spanned by (1,2,-3) and (0,-4,2). Diagonalize A.

Answer 1

To diagonalize the reflection matrix A, find and normalize the cross product of the plane's spanning vectors, construct an orthonormal basis, form the matrix Q with these vectors, create a diagonal matrix D, and multiply QDQ^T.

Step-by-step explanation:

To diagonalize the matrix A representing a reflection through a plane spanned by (1,2,-3) and (0,-4,2), we must find an orthogonal basis for the plane and then include a vector normal to the plane. In this case, the cross product of the spanning vectors gives us the normal vector. Once we have the orthogonal basis consisting of the two spanning vectors and the normal vector, we normalize them to get an orthonormal basis. After that, we construct a matrix Q with these orthonormal vectors as its columns.

The reflection matrix A can be represented in the form A = QDQT, where D is a diagonal matrix with entries 1 for the directions along the plane and -1 for the direction normal to the plane. The matrix QT will then be used to transform original coordinates into the coordinates of the orthonormal basis, apply the reflection as given by D, and then Q will be used to transform the coordinates back to the original basis. This process essentially diagonalizes the reflection matrix A.

To summarize the actual calculation steps:

1. Compute the cross product of the spanning vectors to find the normal vector.
2. Normalize the spanning vectors and the normal vector to get an orthonormal basis (v1, v2, v3).
3. Form the matrix Q with v1, v2, and v3 as its columns.
4. Create the diagonal matrix D with entries (1, 1, -1).
5. Multiply QDQT to get the diagonalized reflection matrix A.