TITLE: Calculating Eigenvectors: Is my book wrong?
QUESTION [7 upvotes]: I have a covariance matrix:
$$
   S= \begin{pmatrix}
    16 & 10 \\
    10 & 25
    \end{pmatrix}
$$
I calculate my eigenvalues correctly (the same as what the book finds);
$\lambda_1 = 31.47$ , $\lambda_2 = 9.53$
But now it comes to calculating eigenvectors:
I do everything as I was taught way back in Elementary Linear Algebra.

$S X = \lambda v$ {where v is the eigenvector}
$(S - I \lambda)v$
Get Row-Echelon Form

But when I do this I get the following reduced matrix:
$$
    \begin{pmatrix}
    1 & -.646412 & 0 \\
    0 & 0 &0 
    \end{pmatrix}
$$
But this result doesn't seem consistent with my textbook which says that the eigenvectors are;
$(0.54 , 0.84)^T$ and $(0.84 , -0.54)$
I looked online for calculators and found one consistent with the book and a few consistent with my result:
Consistent with Book: http://comnuan.com/cmnn01002/
Consistent with Me: http://www.arndt-bruenner.de/mathe/scripts/engl_eigenwert2.htm
Any ideas?
Additional Information:

This problem stems from Principal Component Analysis

REPLY [2 votes]: $(0.54,0.84) = (9,14) \cdot 0.06,$ and $9/14 \approx 0.642857\ldots,$ so I wonder if you get $0.646412$ by rounding too early, before dividing. Maybe your reduced matrix should have been $\left[ \begin{array}{ccc} 1 & -0.642857\ldots & 0 \\ 0 & 0 & 0 \end{array} \right],$ which would be consistent with the answer being $(0.54,0.84).$