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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 Z_{1}, Z_{2}, .... , Z_{n} 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 Z_{1}, Z_{2}, .... , Z_{n}. Let vector X represent the coordinates of a point expressed relative
to the usual E-basis and let vector X_{Z} represent the coordinates of the same point expressed
relative to the Z-basis (eigenbasis). Then

X = ZX_{Z}

and

X_{Z} = 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 X_{Z} = Z^{ -1 }X where the vector X
is expressed relative to the Z-basis (eigenbasis) as X_{Z} .

2. A linear point transformation effected by the diagonal matrix D in the Eigenvector Coordinate
System. This point transformation is given by U = DX_{Z} . The effect of this transformation is
simply stretching (or compressing) effects directed in the directions of the different coordinate
system axes Z_{1}, Z_{2}, .... , Z_{n} with magnitudes given by the eigenvalues k_{1}, k_{2}, .... , k_{n}.

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 }

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