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On page 301 of https://stanford.edu/~dntse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter7.pdf, equation (7.33) says that the magnitude of the dot product of two unit complex vectors is equal to the magnitude of the cosine angle between them.

However, from sources online, I've found that if $\vec a, \vec b$ are unit complex vectors then $\cos (\theta) = Re(\vec a \cdot \vec b)$, which is clearly not the same as (7.33). Is this a mistake in the book?

Furthermore, on the same page 301, it is stated that "The conditioning of $H$ is determined by how aligned the spatial signatures of the two transmit antennas are: the less aligned the spatial signatures are, the better the conditioning of $H$". Here, "conditioning" refers to the condition number of a matrix $H$ with two columns $\vec h_1, \vec h_2$. Is this statement a heuristic or is there a theorem associated with this statement?

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However, from sources online, I've found that if $\vec a, \vec b$ are unit complex vectors then $\cos (\theta) = Re(\vec a \cdot \vec b)$, which is clearly not the same as (7.33). Is this a mistake in the book?

That is Euclidean angle, whereas the book refers to Hermitian angle $$\cos(\theta) = \left|\left\langle a,b\right\rangle \right|/\left(\left\Vert a\right\Vert \left\Vert b\right\Vert \right)$$

to characterize the complex projection.

Furthermore, on the same page 301, it is stated that "The conditioning of $H$ is determined by how aligned the spatial signatures of the two transmit antennas are: the less aligned the spatial signatures are, the better the conditioning of $H$". Here, "conditioning" refers to the condition number of a matrix $H$ with two columns $\vec h_1, \vec > h_2$. Is this statement a heuristic or is there a theorem associated with this statement?

Equation (7.38) of the book proves the statement, doesn't it?

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  • $\begingroup$ Thanks. Actually equation 7.38 is a special case where $a_1 = a_2 = a$ but even when they are not equal, you can calculate the condition number by solving for eigenvalues and see when $|cos\theta|$ is close to $1$ then it's ill-conditioned $\endgroup$ Commented Jan 17, 2024 at 1:09

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