Usage of the word orthogonal outside of mathematics I always found the use of orthogonal outside of mathematics to confuse conversation You might imagine two orthogonal lines or topics intersecting perfecting and deriving meaning from that symbolize
orthogonal vs orthonormal matrices - what are simplest possible . . . Sets of vectors are orthogonal or orthonormal There is no such thing as an orthonormal matrix An orthogonal matrix is a square matrix whose columns (or rows) form an orthonormal basis The terminology is unfortunate, but it is what it is
Are all eigenvectors, of any matrix, always orthogonal? 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 eigenvectors corresponding to distinct eigenvalues are always orthogonal
Eigenvectors of real symmetric matrices are orthogonal Now find an orthonormal basis for each eigenspace; since the eigenspaces are mutually orthogonal, these vectors together give an orthonormal subset of $\mathbb {R}^n$ Finally, since symmetric matrices are diagonalizable, this set will be a basis (just count dimensions) The result you want now follows
What does it mean for two matrices to be orthogonal? The term "orthogonal matrix" probably comes from the fact that such a transformation preserves orthogonality of vectors (but note that this property does not completely define the orthogonal transformations; you additionally need that the length is not changed either; that is, an orthonormal basis is mapped to another orthonormal basis)