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vaieri [72.5K]
3 years ago
6

Find all values of the angle θ (in radians, with 0 ≤ θ < 2π) for which the matrix a = cos θ −sin θ sin θ cos θ has real eigen

values. (enter your answers as a comma-separated list.)
Mathematics
1 answer:
harina [27]3 years ago
4 0

The matrix

A=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}

has eigenvalues \lambda such that

\det(A-\lambda I)=\begin{vmatrix}\cos\theta-\lambda&-\sin\theta\\\sin\theta&\cos\theta-\lambda\end{vmatrix}=0

(\cos\theta-\lambda)^2+\sin^2\theta=0

(\cos\theta-\lambda)^2=-\sin^2\theta

\cos\theta-\lambda=\pm\sqrt{-\sin^2\theta}

\lambda=\cos\theta\pm\sqrt{-\sin^2\theta}

\sin^2\theta\ge0 for all values of \theta, so we need to have \sin\theta=0 in order for \lambda to be real-valued. This happens for

\sin\theta=0\implies\theta=n\pi

where n is any integer, and over the given interval we have \theta=0 and \theta=\pi.

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