Eigenvector Basis

Examples:

• decomposing motion along the fundamental modes;
• writing solid body motion along the principal axes;
• separation of variables;
• improving numerical schemes;
• ...

Diagonalization:

If we use the eigenvectors of a matrix A as a new basis, so that the transformation matrix P contains the eigenvectors:

then the transformed matrix A' is much simpler than the original A. In particular, it is diagonal:

Reason: for any arbitrary vector

then

So A increases the first coordinate in the eigenvector basis by , the second by , etcetera. That is exactly what the diagonal matrix A' does with the vector of coefficients .

Remember that the relationship between A and A' is

Note: If an matrix A has less than n independent eigenvectors, it is not diagonalizable. It is called defective. Most matrices are however diagonalizable:

• As long as all n-eigenvalues are distinct, the matrix is diagonalizable.
• Normal matrices, which commute with their transpose, AA^H=A^HA, always have a complete set of orthonormal eigenvectors anyway.
• Even if the matrix has less than n different eigenvalues and is not normal, it might still be diagonalizable.