How do you minimize the following quadratic function over all (real) :
Also, I have to somehow write the function in the form of:
in order to minimize it. How do I find , a constant vector space?
How do you minimize the following quadratic function over all (real) :
Also, I have to somehow write the function in the form of:
in order to minimize it. How do I find , a constant vector space?
I want to learn how to do it too. But I am not finding any references for this... Can you tell me the general method to minimize functions using linear algebra as given in your text book. Or at least tell me a good reference?
And ya... I can write 4x_1^2-2x_1x_2+3x_2^2+3x_1-2x_2+1 in the matrix form... I figured out K,f and c.
Hello !
This is Gauss's method.
The general idea of minimizing (I don't think I'm misunderstanding) is to get a sum of linearly independent squares.
The thing is to group the terms containing (for example, but we could start with another one) in a first time, and then, to complete the square or use a known formula. I'll try to show it (and detail the calculations, so don't worry ^^)
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Let's deal with the parenthesis thing. It's here where there are two methods, but one looks more familiar (completing the square) whereas the other one is more thouroughly (using a known formula). I'll show only one;
Complete the square
We need to factor 4 in the blue part because and contains a 2.
-->
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And now, do the same.
Remember one thing : the minimized form is , where is not necessarily 1 or positive. It can be negative and it can be different from 1. Why ? Because we will consider the coordinates in the squares as coordinates in a new basis. And one knows that it is possible to multiply a vector by a scalar, it doesn't change anything.
It may look messy, but hmmm... this one looks sadistic to me lol ! Otherwise, it's quite useful once you've mastered it
we have where and note that K is always chosen to be
symmetric. next we find the eigenvalues of K, which are the roots of so
the eigenvectors of K are: we have now define:
where then , and
now let where then:
which after completing the square gives us:
note that it's clear now that:
Note: the columns of are the normalized (i.e. of length 1) eigenvectors of K. we know that P is orthogonal, so
so, in general, in order to make sure that the determinant of P is 1, you might need to choose the first column of P to be
other than that's not the case for our problem though!
i already gave you the matrix it's obvious that you need to choose: andAlso, I have to somehow write the function in the form of:
in order to minimize it. How do I find , a constant vector space?