Solving for a matrix finding the unique sym matrix (Regression Analys)

[TeenieBopper]

Homework Statement

A)Determine the elements of the matrix G such that GY = (Y1 +Y2 + Y3, Y3-Y1, 0.5Y1 - 0.5Y3 +2Y2)'
B)Find the unique symmetric matrix A such that Y'AY = Y'GY.

Y=
[2
7
6]

Homework Equations

The Attempt at a Solution

I know that Y1=2, Y2=7, and Y3=6. That means the right hand matrix should be

Z=
[15
4
12]

So:
GY=Z
GYY'=ZY'
G=ZY'(YY')^1

Using proc IML in SAS I get something like:
0.3370787 1.1797753 1.011236
0.0898876 0.3146067 0.2696629
0.2696629 0.9438202 0.8089888

However, I was talking to a friend, and he just matched up the coefficients and got
G=
1 1 1
-1 0 1
.5 2 -.5

This also makes sense to me, so I don't understand why our numbers would be so different. Did I do something wrong in my equation above?

For part B, I did the same thing

Y'AY = Y'GY
Y'AYY'=Y'GYY'
Y'AYY'(YY')^-1 = Y'GYY'(YY')^-1
Y'A=Y'G
YY'A=YY'G
(YY')^-1YY'A=YY')^-1YY'G
A=G

However, G is not symmetric (in either case). I'm not sure what else I can do.

Thanks in advance for the help.

 

[mighty2000]

First off, it seems like you have made a small mistake in your first attempt at finding the matrix G. In the equation GYY' = ZY', you have multiplied the matrices in the wrong order. It should be G = ZY'(YY')^-1.

As for your friend's solution, it is also correct. Both of your solutions are valid, as there are multiple matrices that can satisfy the given equation. The important thing is that the resulting matrix G should produce the same output as the given values for Y.

For part B, you have correctly determined that A = G. However, as you mentioned, G is not symmetric. This means that there is no unique symmetric matrix A that can satisfy the equation Y'AY = Y'GY. In this case, you would need to specify whether you want a symmetric matrix A or any matrix that satisfies the equation.

I hope this helps clarify any confusion. Keep up the good work in your studies!