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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?

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  • $\begingroup$ I don't think all those "the" should be there. However, you can find a Jordan basis and orthogonalize it. $\endgroup$ Commented Dec 16, 2020 at 8:28

1 Answer 1

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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 $\vec x = \pmatrix{1\\-1}$.

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