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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,
AAH=AHA, 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.
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