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weetabixharry
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I'm having difficulty understanding the nature of a plane in complex space.
Specifically, I have two complex [itex] (N \times 1) [/itex] vectors, [itex]\underline{u}_1 [/itex] and [itex]\underline{u}_2 [/itex], which are orthonormal:
[tex]\underline{u}_1^H\underline{u}_2 = 0 \\
\Vert \underline{u}_k \Vert = 1 \ \ \ (k=1,2)[/tex]
So, [itex]\underline{u}_1 [/itex] and [itex]\underline{u}_2 [/itex] span a 2D subspace (plane) and any vector in that plane can be written in the form:
[tex]\underline{v} = a_1\underline{u}_1 + a_2\underline{u}_2[/tex]
(for some complex scalars [itex]a_1[/itex] and [itex]a_2[/itex]). What I want to know is whether certain properties of [itex]a_1[/itex] and [itex]a_2[/itex] can tell us something about the nature of [itex]\underline{v}[/itex] (in the context of [itex]\underline{u}_1 [/itex] and [itex]\underline{u}_2 [/itex]). Specifically, what if [itex]a_1, a_2[/itex] are purely imaginary (i.e. [itex]\Re e\{a_1\}=\Re e\{a_2\}=0[/itex])? I feel that such a construction must have specific (visualisable/intuitive) properties, but I can't see what they might be. Can anyone shed any light on this?
Specifically, I have two complex [itex] (N \times 1) [/itex] vectors, [itex]\underline{u}_1 [/itex] and [itex]\underline{u}_2 [/itex], which are orthonormal:
[tex]\underline{u}_1^H\underline{u}_2 = 0 \\
\Vert \underline{u}_k \Vert = 1 \ \ \ (k=1,2)[/tex]
So, [itex]\underline{u}_1 [/itex] and [itex]\underline{u}_2 [/itex] span a 2D subspace (plane) and any vector in that plane can be written in the form:
[tex]\underline{v} = a_1\underline{u}_1 + a_2\underline{u}_2[/tex]
(for some complex scalars [itex]a_1[/itex] and [itex]a_2[/itex]). What I want to know is whether certain properties of [itex]a_1[/itex] and [itex]a_2[/itex] can tell us something about the nature of [itex]\underline{v}[/itex] (in the context of [itex]\underline{u}_1 [/itex] and [itex]\underline{u}_2 [/itex]). Specifically, what if [itex]a_1, a_2[/itex] are purely imaginary (i.e. [itex]\Re e\{a_1\}=\Re e\{a_2\}=0[/itex])? I feel that such a construction must have specific (visualisable/intuitive) properties, but I can't see what they might be. Can anyone shed any light on this?
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