a) Let v1 = (1, 1, 0) and v2 = (1, 0, 1). If the eigenvalue of v1 is 2, what is the eigenvalue of v2?
- I found the answer to this by proving that these eigenvectors are not orthogonal by computing their dot product which turned out to be 1. So their eigenvalues are not distinct and therefore eigenvalue of v1 equals the eigenvalue of v2.
However, I am stumbled upon this question:
b) Find an orthogonal matrix that diagonalises the Matrix A. I found the third eigenvector by simply taking the crossproduct of v1 and v2. So v3 = (1, -1, -1). So I have now have the symmetric matrix A. However, I am unsure on how to progress. Any hints?
You're almost there. Now you just have to replace (v1, v2, v3) by an orthonormal basis of eigenvectors. The matrix with columns composed of these eigenvectors is the one you want.