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Eigenvector Basis

Among all similar matrices (defined by the similarity transformation defined by Eq. 10-13), the simplest matrix is the diagonal one. In the coordinate system where the similar matrix is diagonal, its diagonal entries are the eigenvalues. The question remains, ``what is the coordinate transformation that takes the matrix into its diagonal form?''

The coordinate system is called the eigenbasis or principal axis system, and the transformation that takes it there is particularly simple.

The matrix that transforms from a general (old) coordinate system to a diagonalized matrix (in the new coordinate system) is the matrix of columns of the eigenvectors. The first column corresponds to the first eigenvalue on the diagonal matrix, and the $n^{th}$ column is the eigenvector corresponding the $n^{th}$ eigenvalue.

$$\left( \begin{array}{c} \text{The}\ \text{Diagonalized}\ \text{... ...rray}{c} \text{Eigenvector}\ \text{Column}\ \text{Matrix} \end{array} \right)$$ (10-29)

This method provides a method for finding the simplest quadratic form.