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?

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- Jul 7th 2008, 03:11 PMJCS007How to minimize a quadratic function using linear algebra
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? - Jul 7th 2008, 09:20 PMIsomorphism
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. - Jul 7th 2008, 10:38 PMniniandcat
i just need the answer too ,who can help us?

- Jul 8th 2008, 12:41 AMMoo
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 ^^)

------------------------------

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.

-->

-------------------------------

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 (Tongueout) - Jul 8th 2008, 01:26 AMNonCommAlg
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!

Quote:

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?