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

• TeenieBopper
In summary, there are multiple matrices that can satisfy the given equations, and both your solution and your friend's solution are valid. However, there is no unique symmetric matrix A that can satisfy the equation in part B.
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]

## 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.

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!

## 1. What is a matrix in regression analysis?

A matrix in regression analysis is a table of data that represents the relationship between variables. It is used to solve for the unique sym matrix, which is a symmetric matrix of coefficients that is used to calculate the regression equation.

## 2. How is the unique sym matrix calculated in regression analysis?

The unique sym matrix is calculated by using the Ordinary Least Squares (OLS) method, which minimizes the sum of squared errors between the predicted values and the actual values in the data set.

## 3. What is the purpose of finding the unique sym matrix in regression analysis?

The unique sym matrix is used to calculate the regression equation, which is a line of best fit that can be used to make predictions about the relationship between variables. It is an important tool in understanding and analyzing data in regression analysis.

## 4. How is the unique sym matrix different from other matrices in regression analysis?

The unique sym matrix is different from other matrices in regression analysis because it is a symmetric matrix, meaning that it is equal to its transpose. This property allows for easier calculations and interpretation of the results.

## 5. What are some common applications of solving for the unique sym matrix in regression analysis?

Solving for the unique sym matrix is commonly used in fields such as statistics, economics, and social sciences to analyze and interpret data. It can also be used in predictive modeling, forecasting, and trend analysis to make informed decisions based on the relationships between variables.