# Using Principal Component Analysis Based on Sample Covariance? [But really how to get these eigenvalues?] [closed]

I have the following sample covariance matrix;

$$S= \begin{pmatrix} 16 & 10 \\ 10 & 25 \end{pmatrix}$$

Now to use PCA I know I need to find the eigenvalues and eigenvectors. I know how to get the eigenvalues which gives me:

$$\lambda_1 = 31.47 , \lambda_2 = 9.53$$

But how do I get the following eigenvectors?

$(0.54, 0.84)^T$ and $(0.84 , -0.54)$

Can someone do it step by step for me? I thought I knew how to find eigenvalues by, hand but I've always used whole numbers when learning eigenvectors.

EDIT: This is one of the earlier steps to problems involving Principal Component Analysis. My book says, "here are the eigenvalues" and I can confirm that they are the eigenvalues but I have 0 clue how they calculated the eigenvectors. Perhaps I misunderstood PCA?

• Feel free to add the self-study tag – Ferdi Nov 22 '17 at 23:21
• It seems $(S− \lambda I)x = 0$ has no solution after we plug in eigenvales $\frac{41\pm \sqrt{481}}{2}$, Don't know how this is handled numerically – Deep North Nov 23 '17 at 6:00
• That's what I was thinking. I wonder if I misunderstood PCA. I thought we just took eigenvectors for the covariance matrix. I'll add a correction if I misunderstood. Otherwise I was able to get eigenvectors but they're not what the book has. – Nicklovn Nov 24 '17 at 17:48
• See en.wikipedia.org/wiki/…. @Deep If you could find no solution, you must have erred in your calculation. – whuber Nov 24 '17 at 18:19
• Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. – gung - Reinstate Monica Nov 25 '17 at 0:09