# 3x3 matrix Eigenvector values does not match

I'm trying to solve a matrix $$Q$$ (as shown in the SCREENSHOT) to find its eigenvectors.

The solutions are provided in the book directly and I was trying to solve it by hand but I cannot match my answers with the given solution.

Moreover I cannot make sense of where did $$\sqrt{1/2}$$ come from. I even tried solving the problem using an online matrix calculator such as this.

Am I missing something?

• Welcome to MSE. In the future please learn to use MathJax to properly format math expressions. – Lee David Chung Lin Apr 20 at 22:30
• when the matrix is symmetric and real, it is possible to find $P^T Q P = D,$ where $P^T P = I.$ Here $Q$ emphasizes that you have a quadratic form. – Will Jagy Apr 20 at 22:37
• I did one of these yesterday, where I did not bother looking for orthogonal, just nonsingular $P$ with $P^T H P = D.$ This is the correct change of variable for quadratic forms. math.stackexchange.com/questions/3193138/… – Will Jagy Apr 20 at 22:45
• Well, your question is missing your work and in particular the eigenvectors that you’ve worked out. Remember, though, that there’s no such thing as the eigenvectors: any nonzero scalar multiples of them will do. The scale factors were probably chosen to make some computation down the line produce a “nice” answer, but we can only guess since you don’t provide any of that. – amd Apr 20 at 22:52