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I've spent an incredible amount of time on this question, taken out of a library rau textbook. It tests all parts of my understanding of matrix and vector knowledge. I cannot do it!
Basically, min (b (vector)) (y - Xb)^t(y-Xb) ; y and b are vectors. X is a matrix. This is the general least-squares problem, when X is an n*k matrix.
The critical point of b* ( I proved this earlier, had some luck with that one) is (X^TX)^-1X^ty (^t = transpose, ^-1 = inverse)
Now, I must substitute my b* into the original equation, to try and find the minimum value function.
I've tried both methods; multiplying out the initial (1st) expression until it's in its most basic form, then substituting b* in. I also tried substituting b* in from the start, and couldn't do that either.
Please could someone help me with this problem, I am going crazy.
I'm not used to using Latex so I apologise for the way I formatted my question. I tried both methods and really do not know how to get to the answer;
At one stage, (the first method I tried, expanding before substituting) - I got the expression y^ty, am I right in assuming y^ty=yy^t = I?
I have the answer, if you think it'd be useful, just ask.
I did try and cheat, and work backwards, the thing is I am so close. I just can't get the identity matrix in both sides....
If you could try writing out the problem and substituting in b and working through it, i'm here to discuss it because it's really challenging
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