Here is an example that may be helpful.
A <- matrix(c(-3,5,4,-2),nrow=2,byrow=TRUE)
eigs <- eigen(A)
eigs
$values
[1] -7 2
$vectors
[,1] [,2]
[1,] -0.7808688 -0.7071068
[2,] 0.6246950 -0.7071068
The eigenvectors may be scaled differently because they are not unique
(or have a different sign), but Ax = lambda x, for an eigenvalue
lambda.
#Ax
A %*% eigs$vectors[,1]
[,1]
[1,] 5.466082
[2,] -4.372865
# λx
> -7 * eigs$vectors[,1,drop=FALSE]
[,1]
[1,] 5.466082
[2,] -4.372865
The eigenvectors for proc iml are scaled similarly, but have different signs.
proc iml;
A = {-3 5, 4 -2};
print A;
eigval = eigval(A);
evec = eigvec(A);
print eigval;
print evec;
A_x = A*evec[,1];
lambda_x = eigval[1,1]*evec[,1];
print A_x;
print lambda_x;
quit;
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