For the sake of brevity, I'm given a 3x3 symmetric matrix with real entries with no further information as to what the rows and columns encode. (eg. this not need to be the case but the columns may very well stand for i, j and k vectors and the rows for the transformed i, j and k)
Suppose we didn't know this and this matrix acts on a vector space spanned by an arbitrary basis. What is required of this starting basis for the eigendecomposition to work?
For example, when looking at the above proof, it uses the fact that a dot product may be written as a (in this case) 1x3 row vector multiplying a 3x1 column vector. But this not need to be the case, right? Unless I gravely misunderstood this is only true for Cartesian coordinates. If the starting basis were not orthogonal, then dotting a pair of vectors in that basis will produce crossed terms and this does not reflect a row vector multiplying a column vector.
So much for brevity. Generally speaking, what assumptions need to be made when performing eigendecomposition on a real symmetric matrix?
edit: It seems I've been vague so I've added a little example to better illustrate my point.
eigendecomposition of a real symmetric matrix is based on the idea that eigenvectors from different eigenspaces are orthogonal with respect to the dot product. I have a problem with this statement as it seems to make the assumption that we're working in cartesian coordinates. Is this true or is there a fault in my reasoning?
In the above example the alphas are certainly orthogonal if the u's are like i and j. However if it's not specified, do we have to make this assumption?