An orthonormal set must be linearly independent, and so it is a vector basis for the space it spans. Such a basis is called an orthonormal basis. ... Another instance when orthonormal bases arise is as a set of eigenvectors for a symmetric matrix.

Are all orthogonal vectors eigenvectors?

In general, for any matrix, the eigenvectors are NOT always orthogonal. But for a special type of matrix, symmetric matrix, the eigenvalues are always real and the corresponding eigenvectors are always orthogonal.

Do eigenvectors always form a basis?

No, of course not. For example, (0100) has 0 as its only eigenvalue, with eigenspace (x0). Thus there are not enough independent eigenvectors to form a basis.