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If an operator $$T\colon R^3\to R^3$$ is presented,with respect to an orthonormal basis,by a matrix

$ A= \begin{bmatrix} 0&1&0\\1&0&0\\0&0&1 \end{bmatrix} $. What is the corresponding spectral family?

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1 Answer 1

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Let $M = \left[ \matrix{-1/\sqrt{2} & 1/\sqrt{2} & 0 \\ 1/\sqrt{2} & 1/\sqrt{2} & 0 \\ 0 & 0 & 1} \right]$. Note that $M = M^T = M^{-1}$, and $MAM= \left[ \matrix{-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1} \right]$. Thus, the eigenvalues of $A$ are $-1$ and $1$, and the projectors can be calculated by observing that $ ME_{-1}M = \left[ \matrix{ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0} \right] $ and $E_1 = I$.

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