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## Main Question or Discussion Point

Hey all, I'm trying to understand the Capon Method for some reading into MIMO radar [1] and went to the original source [2] trying to wrap my head around the linear algebra involved. From [2], the Capon method is derived in Section 5.4.1, which refers to Result R35 in Appendix A. It is about finding a unique solution to the following minimization problem:

[itex] min_{X} X^*AX [/itex] subject to [itex]X^*B=C [/itex]

It's been a while since I took linear algebra, but I remember most of the basics. However, I can't wrap my head around the line after Proof's equation A.9.6:

[tex] \Delta^*AX_0 = \Delta^*B(B^*A^{-1}B)^{-1}C^*=0 [/tex]

It seems to come out of nowhere to me, and I'm sure it would require more context which is why I provided the source so it wouldn't be as hard to read this post.

My attempts to work it out on scratch paper haven't been fruitful, mainly because I've tried starting with the equation [itex]X^*B=C[/itex] and doing stuff like this:

[tex]X_0^*B=C \Rightarrow B^*X_0=C^* \Rightarrow X_0 = B^{*-1}C^* [/tex]

However, since [itex]B[/itex] is not square, I don't think I can have [itex]B^{*-1}[/itex]. Without the ability to do things like that, I don't know how to systematically get to the equation above (line after A.9.6). It feels like you'd have to jump multiple steps to get the part of the equation that's inverted and inside parenthesis, for example.

Any advice would be much appreciated, I'd really like to understand how this works.

Thank you!

References:

[1]: MIMO Radar Signal Processing by Jian Li and Petre Stoica.

[2]: Spectral Analysis of Signals by Petre Stoica and Randolph Moses.

Link: http://user.it.uu.se/~ps/SAS-new.pdf

[itex] min_{X} X^*AX [/itex] subject to [itex]X^*B=C [/itex]

It's been a while since I took linear algebra, but I remember most of the basics. However, I can't wrap my head around the line after Proof's equation A.9.6:

[tex] \Delta^*AX_0 = \Delta^*B(B^*A^{-1}B)^{-1}C^*=0 [/tex]

It seems to come out of nowhere to me, and I'm sure it would require more context which is why I provided the source so it wouldn't be as hard to read this post.

My attempts to work it out on scratch paper haven't been fruitful, mainly because I've tried starting with the equation [itex]X^*B=C[/itex] and doing stuff like this:

[tex]X_0^*B=C \Rightarrow B^*X_0=C^* \Rightarrow X_0 = B^{*-1}C^* [/tex]

However, since [itex]B[/itex] is not square, I don't think I can have [itex]B^{*-1}[/itex]. Without the ability to do things like that, I don't know how to systematically get to the equation above (line after A.9.6). It feels like you'd have to jump multiple steps to get the part of the equation that's inverted and inside parenthesis, for example.

Any advice would be much appreciated, I'd really like to understand how this works.

Thank you!

References:

[1]: MIMO Radar Signal Processing by Jian Li and Petre Stoica.

[2]: Spectral Analysis of Signals by Petre Stoica and Randolph Moses.

Link: http://user.it.uu.se/~ps/SAS-new.pdf