Eigenvalues:
There is a single root: and a double rootEigenvectors corresponding to satisfy
Solving using Gaussian elimination:
Equation (2') gives v1y = -v1z and then (1') gives v1x = 2 v1z.The general solution space is:
We choose v1z=1 to get
Eigenvectors corresponding to satisfy
Solving using Gaussian elimination:
Equation (1') gives v2x = v2y + v2z. There are two unknown parameters.The general solution space is:
We need two independent eigenvectors to span the space corresponding to this multiple root.We can use the two vectors above, which means choosing v2y=1 and v2z=0 for one, and v2y=0 and v2z=1 for the other. That gives
If the three vectors , , and are used as basis, A becomes diagonal. So despite the multiple root, this A is still diagonalizable. But if the solution space for the second eigenvalue would have been one-dimensional, the matrix would not have been diagonalizable.
y