Question

Verify the spectral theorem for each of the following symmetric matrices by finding an orthonormal basis of the appropriate vector space, the change of basis matrix to this basis, and the spectral decomposition.

a) A = [3 -1][-1 5]
E- = Span{(-√2/2, √2/2)} and Es = Span{(√2/2, √2/2)}. The basis for R2 is {(-√2/2, √2/2), (√2/2, √2/2)}, which is an orthonormal basis.
0 = [-√2/2 √2/2] [√2/2 √2/2]
A = [-1√2/2 + 5√2/2][√2/2 √2/2]
Solution is given. No need to solve the problem completely. Explain how to find Pe- and Pes. Please don't find the eigenvector and orthonormal basis.

Answer 1

The spectral theorem is verified using the projection matrices Pe- and Pes, constructed using the outer products of the orthonormal basis vectors for the eigenspaces E- and Es of a symmetric matrix.

Step-by-step explanation:

To verify the spectral theorem for the given symmetric matrix, we work with the provided orthonormal basis of R2, which consists of the vectors (-√2/2, √2/2) and (√2/2, √2/2). While the question specifies that we should not find the eigenvectors and orthonormal basis, it does ask for an explanation of how to find Pe- and Pes, which are the projection matrices corresponding to the eigenspaces E- and Es respectively.

To find these projection matrices, we construct them using the outer products of the basis vectors of the eigenspaces. For instance, the projection matrix for an eigenspace spanned by a unit vector u is given by the outer product uuT. In general, if the eigenspace has more than one basis vector, the projection matrix would be the sum of the outer products of those basis vectors with themselves.

Once we have the projection matrices, the spectral decomposition of the matrix A can be expressed as the sum of the outer products of each eigenvector with itself multiplied by the corresponding eigenvalue. This decomposition results in a diagonal matrix when our change of basis matrix is the one formed by the eigenvectors.