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A linear point transformation y = ax viewed as occurring in three steps

Let A be an nxn matrix all of whose eigenvalues are distinct.. Then A is similar to a diagonal matrix D whose diagonal elements are these eigenvalues. In addition, all of the n eigenvectors are linearly independent. Let Z1, Z2, .... , Zn be the normalized eigenvectors and let them constitute a basis for an Eigenvector Coordinate System (an oblique coordinate system centered at the origin called the “eigenbasis”). Let Z be a matrix whose columns contain the normalized eigenvectors Z1, Z2, .... , Zn. Let vector X represent the coordinates of a point expressed relative to the usual E-basis and let vector XZ represent the coordinates of the same point expressed relative to the Z-basis (eigenbasis). Then

X = ZXZ

and

XZ = Z -1 X

The linear point transformation effected by matrix A is that same one effected by the diagonal matrix D in the Eigenvector Coordinate System. It is just viewed from a different coordinate system. The linear point transformation effected by matrix A can be viewed as occurring in three steps:

1. A change of basis from the usual coordinate system (E-basis) to the Eigenvector Coordinate System (the canonical coordinate system where the actual point transformation is performed). This change of basis corresponds to the coordinate transformation XZ = Z -1 X where the vector X is expressed relative to the Z-basis (eigenbasis) as XZ .

2. A linear point transformation effected by the diagonal matrix D in the Eigenvector Coordinate System. This point transformation is given by U = DXZ . The effect of this transformation is simply stretching (or compressing) effects directed in the directions of the different coordinate system axes Z1, Z2, .... , Zn with magnitudes given by the eigenvalues k1, k2, .... , kn.

3. A change back to to the original E-basis given by Y = ZU .

Putting these three steps together we have

Y = ZD Z -1 X

thus

A = ZD Z -1