# Matricial equation problem

• Feb 13th 2013, 09:52 AM
rud25
Matricial equation problem
Well I hope I'm posting this in the right place.

I have a problem that is written in the following form:
(Ax - y)' * (Ax - y) where the symbol ( ' ) denotes the transpose of a matrix/vector.

I have to arrange this equation and put in a form where I have:

x'*Q*x

The first statement is from a minimization problem in quadratic programming, so that's why I have to put it in the form of the second statement, so my problem here is really rearranging the first statement into the second statement.

My regards
• Feb 13th 2013, 01:02 PM
DavidB
Re: Matricial equation problem
I don't know if there is a general rule for simplifying the original expression: ([A]x - y)T * ([A]x - y)

However, if it is supposed to be capable of being rearranged into the form for Quadratic Programming, the [Q] matrix is symmetric. Not sure if that helps to work backward.
Do you know ahead of time if the [A] matrix has any special properties?
• Feb 14th 2013, 03:54 AM
rud25
Re: Matricial equation problem
Quote:

Originally Posted by DavidB
I don't know if there is a general rule for simplifying the original expression: ([A]x - y)T * ([A]x - y)

However, if it is supposed to be capable of being rearranged into the form for Quadratic Programming, the [Q] matrix is symmetric. Not sure if that helps to work backward.
Do you know ahead of time if the [A] matrix has any special properties?

I know [A] from my problem. It is nonsquare and has generally more columns than rows.

I am using the MATLAB routine quadprog to solve my quadratic problem, so that's why I need to know the matrix [Q] which is symmetric as you well said.
Generally I have a much simple problem to solve which is:

minimize 1/2*norm(x)^2 (I'm referring to the norm p=2 here).

In this case I just have to: norm(x)^2 = x'*x, and so the [Q] matrix for this problem is the identity matrix: x'*[I]*x

The problem I posted yesterday is a special case of the problem I just described. In this case I have the problem:
min 1/2*norm(A*x-y)^2. Then I tried to follow the same strategy as before:

norm(A*x-y)^2 = (Ax - x)' * (Ax - y)...

Maybe there is a simpler strategy to solve this minimization problem...