Let formula_128 be an eigendecomposition of formula_2, where formula_130 is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of formula_2, and formula_132 is a "real" diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix formula_2 may be regarded as a diagonal matrix formula_132 that has been re-expressed in coordinates of the (eigenvectors) basis formula_130. Put differently, applying "M" to some vector z in our coordinates system ("M"z), is the same as changing the basis of our z to the eigenvector coordinate system using "P"−1 ("P"−1z), applying the stretching transformation "D" to it ("DP"−1z), and then changing the basis back to our system using "P" ("PDP"−1z).