  The paper presents the graph Fourier transform (GFT) of a signal in terms of
its spectral decomposition over the Jordan subspaces of the graph adjacency
matrix $A$. This representation is unique and coordinate free, and it leads to
unambiguous definition of the spectral components ("harmonics") of a graph
signal. This is particularly meaningful when $A$ has repeated eigenvalues, and
it is very useful when $A$ is defective or not diagonalizable (as it may be the
case with directed graphs). Many real world large sparse graphs have defective
adjacency matrices. We present properties of the GFT and show it to satisfy a
generalized Parseval inequality and to admit a total variation ordering of the
spectral components. We express the GFT in terms of spectral projectors and
present an illustrative example for a real world large urban traffic dataset.
