Are eigenvectors returned by R function eigen() wrong?

#eigen values and vectors
a <- matrix(c(2, -1, -1, 2), 2)

eigen(a)

I am trying to find eigenvalues and eigenvectors in R. Function eigen works for eigenvalues but there are errors in eigenvectors values. Is there any way to fix that?

Some paper work tells you

• the eigenvector for eigenvalue 3 is (-s, s) for any non-zero real value s;
• the eigenvector for eigenvalue 1 is (t, t) for any non-zero real value t.

Scaling eigenvectors to unit-length gives

s = ± sqrt(0.5) = ±0.7071068
t = ± sqrt(0.5) = ±0.7071068

Scaling is good because if the matrix is real symmetric, the matrix of eigenvectors is orthonormal, so that its inverse is its transpose. Taking your real symmetric matrix a for example:

a <- matrix(c(2, -1, -1, 2), 2)
#     [,1] [,2]
#[1,]    2   -1
#[2,]   -1    2

E <- eigen(a)

d <- E[[1]]
#[1] 3 1

u <- E[[2]]
#           [,1]       [,2]
#[1,] -0.7071068 -0.7071068
#[2,]  0.7071068 -0.7071068

u %*% diag(d) %*% solve(u)  ## don't do this stupid computation in practice
#     [,1] [,2]
#[1,]    2   -1
#[2,]   -1    2

u %*% diag(d) %*% t(u)      ## don't do this stupid computation in practice
#     [,1] [,2]
#[1,]    2   -1
#[2,]   -1    2

crossprod(u)
#     [,1] [,2]
#[1,]    1    0
#[2,]    0    1

tcrossprod(u)
#     [,1] [,2]
#[1,]    1    0
#[2,]    0    1

---

How to find eigenvectors using textbook method

The textbook method is to solve the homogenous system: (A - λI)x = 0 for the Null Space basis. The NullSpace function in my this answer would be helpful.

## your matrix
a <- matrix(c(2, -1, -1, 2), 2)

## knowing that eigenvalues are 3 and 1

## eigenvector for eigenvalue 3
NullSpace(a - diag(3, nrow(a)))
#     [,1]
#[1,]   -1
#[2,]    1

## eigenvector for eigenvalue 1
NullSpace(a - diag(1, nrow(a)))
#     [,1]
#[1,]    1
#[2,]    1

As you can see, they are not "normalized". By contrasts, pracma::nullspace gives "normalized" eigenvectors, so you get something consistent with the output of eigen (up to possible sign flipping):

library(pracma)

nullspace(a - diag(3, nrow(a)))
#           [,1]
#[1,] -0.7071068
#[2,]  0.7071068

nullspace(a - diag(1, nrow(a)))
#          [,1]
#[1,] 0.7071068
#[2,] 0.7071068