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I came across this problem but do not know how to approach it. Could someone point me in the right direction?

Let $T:\mathbb{R}^4\to\mathbb{R}^4$ be a linear transformation. Then which of the following is true?

(A) $T$ must have some real eigenvalues which may be less than 4 in number.

(B) $T$ may not have any real eigenvalues at all.

(C) $T$ must have infinitely many real eigenvalues.

(D) $T$ must have exactly 4 real eigenvalues.

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Check this and this out. –  dtldarek Dec 3 '12 at 10:47

2 Answers 2

up vote 2 down vote accepted

HINT:

Consider the transformation $T(x)=Ax, \ x \in \mathbb R^4$ where $$A=\begin{pmatrix}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0 \end{pmatrix}.$$

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thanks for the example.I have checked the matrix A and have seen that all eigenvalues are imaginary.So T may not have any real eigenvalue at all. –  learner Dec 3 '12 at 11:29

HINT: Make your life easy and assume $T$ to be diagonal...

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