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               Eigenvectors and their meaning 




If all the n eigenvalues λ1, λ2, .... , λn of an n-square matrix A are distinct, then the n eigenvectors Z1, Z2, .... , Zn are linearly independent and the matrix is similar to a diagonal matrix D. This means that matrix A represents the same linear transformation as the diagonal matrix D – it is simply referred to a different basis (i.e. coordinate system). We know the character of the point transformation effected by a diagonal matrix – it represents simply stretching (or compressing) effects directed in the directions of its different coordinate axes. This stretching (or compressing) effected by the diagonal matrix will occur in a coordinate system whose axes correspond to the eigenvectors Z1, Z2, .... , Zn and with magnitudes given by the eigenvalues λ1, λ2, .... , λn.



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