Find the unitary matrix $U$ and the upper triangular matrix $T$ such that $U^{-1}AU = T$ where $A = \pmatrix{3&2\\-2&-1}$ has the eigenvalue $\lambda = 1$ (twice) corresponding to the eigenvector $\vec x = \pmatrix{1\\-1}$.
Hi guys.
I am stuck in finding unitary matrix.
As I know, the unitary matrix must be invertible but this matrix has eigenveector of multiplicity.
How can I find unitary matrix here?
Hint: The columns of $U$ must form an orthonormal basis of $\Bbb C^2$, and its first column must be a multiple of the eigenvector $\vec x = \pmatrix{1\\-1}$.