- #1

Milliarde

- 2

- 0

[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