Linear algebra and choosing training vector

[perplexabot]

Hey all. So I have been reading this article and have a question I would like to ask. I will be referring to this article extensively so it would be kind of you to open it: http://www.ee.ucr.edu/~yhua/MILCOM_2013_Reprint.pdf

I believe reading the article is not required to answer my questions (i did personally read it tho up until where I am stuck), but I will refer to the equation numbers in the article instead of typing them out here.

So, on page 5 of the pdf, there is a matrix, G₂, or equation number (22). I am basically trying to figure out how they constructed this matrix! Specifically the left two partitions.

• Equation (15) gives the definition of G, and shows that G depends on u(k).
• Equation (12) defines u(k) to be [u₁^T | u₂^T | $\bar{g}$ ^T | 1].

• The two lines after equation (12) define u₁^T and u₂^T
• Equation (5) defines $\bar{g}$ ^T, g_i^T, and g_r^T
• Now that we have all the definitions stated we can go back to equation (22), or G₂. It is stated in the two lines above equation (22) that m = 2 (m is the column size of $\bar{g}$ ^T). Right?

IT IS AT THIS POINT WHERE MY CONFUSION BEGINS

• Now if $\bar{g}$ ^T contains only two elements that means, according to equation (5), g_i^T, and g_r^T would be scalars, right?
• Continuing with this logic and going to the definition of u₂^T, the difference of the Kronecker product of two scalars, would be zero! would it not? Hence, u₂^T = 0
• According to the previous bullet point, this would make the second column/partition (from the left) of equation (22) equal to 0, which contradicts what is shown in equation (22)
• Doing the same with u₁^T and the first column/partition (from the left) of equation (22) would also yield a different answer.

That is my logic on this. I know I am doing something wrong, but I am not sure what it is. Please help me out here as I have been trying to understand this for a quite a bit. Thank you for read and your time : )

 

[perplexabot]

Hey! I don't know if anyone cares but I believe I got it! It was right in front of my face the whole time. So the G₂ I was trying to find was for a real system (not complex), this required the use of equation (21)! I considered that earlier but what I thought was that since equation (21) had only 3 partitions while G₂ had 4 partitions that equation doesn't work. It turns out tho that the first partition of equation (21) provided three elements when solved. These three elements covered the first two partitions of G₂.

Damn, if feels good to figure something out after days of trial and error. Sorry to take your time and thank you for reading. I am still open to comments of course : )