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Homework Statement
Suppose that a matrix A has real entries (which we always assume) and a complex
(non-real) eigenvalue [tex]\lambda[/tex]= a + ib , with b not equal to 0. Let W = U + iV be the corresponding
complex eigenvector, having real and imaginary parts U and V , respectively. Show that
U and V are necessarily linearly independent (meaning that one vector is not a scalar
multiple of the other).
HINT: Argue by contradiction: suppose that U and V are dependent, say, V = rU for
some scalar r, and derive a contradiction (that is, a statement that follows logically from
the supposition, but which is false, such as 0 > 1).
Homework Equations
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The Attempt at a Solution
This question confuses me because I don't even know how to go about it or where to start. How should I do this? Where do I begin??
Thanks!