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For a given $N \times 1$ vector $x$, $x \neq 0$, consider the matrix $R = xx^H$. Identify the range space and null space of $R$, and thus determine its rank. Here $H$ is a Hermitian operator.

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  • $\begingroup$ You can determine the rank from the dimension of the null space alone, using the rank-nullity theorem. $\endgroup$ – Stefan Smith Apr 16 '13 at 23:07
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Hint: For every vector $y$, $Ry=(x^Hy)x$ where $x^Hy$ is a scalar.

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