# Extracting Eigenvalues from A Linear Regression In R

The function below extracts the eigenvalues and an eigenvector from a linear regression model. I wrote this function so that the eigenvalues and vectors are the same as those reported by the SAS `PROC REG`

procedure when using the `COLLIN`

option.

```
scaleEigen <- function(fit){
# Calculate the eigenvalues and eigenvectors
#
# args: fit - an object of the lm or glm class from which you would
# like to extract the X matrix
#
# Output: eigenvalues and eigenvectors of the X'X matrix
#
# Method:
# fit <- lm(y ~ x1 + x2 + x3, data = dataset)
# summary(fit)
# scaledEigen(fit)
#
model.data <- fit$model # The data used in the regression
Xvec <- as.matrix(cbind(1, model.data[,-1])) # X matrix including intercept
Xvec <- apply(Xvec, 2 , function(x) x/sqrt(sum(x^2))) # Rescale the X matrix
eigenValues <- eigen(t(Xvec)%*%Xvec)
return(eigenValues)
}
```

For example, using cars data from the R dataset package we can extract eigenvalues and eigenvectors of a **X'X** matrix which can be used to assess multicollinearity of regressors in a linear regression.

```
data(cars)
cars.lm <- lm(dist ~ speed, data = cars)
summary(cars.lm)
# Extract the eigenvalues and eigenvectors from cars.lm using
# the function scaleEign above
scaleEigen(cars.lm)
```

This produces the eigenvalues and their associated eigenvectors of a class `list`

as shown in the R output below.

```
$values
[1] 1.94680083 0.05319917
$vectors
[,1] [,2]
[1,] -0.7071068 0.7071068
[2,] -0.7071068 -0.7071068
```

You can also assign the output to a an object and access the values using the dollar sign as below:

```
eigenV <- scaleEigen(cars.lm)
eigenV$values # Eigenvalues
eigenV$vectors # Eigenvectors
```

In a regression analysis, if there is one or more small eigenvalues of the **X'X**, matrix it implies there are near-linear relationships (dependencies) among regressors and we may have multicollinearity issues in the analysis.