Editing Matrix similarity (section)

== Motivating example ==

When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in {{math|ℝ<sup>3</sup>}} when the [[axis–angle representation|axis of rotation]] is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were aligned with the positive {{mvar|z}}-axis, then it would simply be
<math display="block">S = \begin{bmatrix}
  \cos\theta & -\sin\theta & 0 \\
  \sin\theta &  \cos\theta & 0 \\
           0 &           0 & 1
\end{bmatrix},</math>
where <math>\theta</math> is the angle of rotation. In the new coordinate system, the transformation would be written as 
<math display="block">y' = Sx',</math>
where {{mvar|x'}} and {{mvar|y'}} are respectively the original and transformed vectors in a new basis containing a vector parallel to the axis of rotation. In the original basis, the transform would be written as
<math display="block">y = Tx,</math>
where vectors {{mvar|x}} and {{mvar|y}} and the unknown transform matrix {{mvar|T}} are in the original basis. To write {{mvar|T}} in terms of the simpler matrix, we use the change-of-basis matrix {{mvar|P}} that transforms {{mvar|x}} and {{mvar|y}} as <math>x' = Px</math> and <math>y' = Py</math>:
<math display="block">\begin{align}
  &            &  y' &= S x'  \\[1.6ex]
  &\Rightarrow & P y &= S P x \\[1.6ex]
  &\Rightarrow &   y &= \left(P^{-1} S P\right) x = T x
\end{align}</math>

Thus, the matrix in the original basis, <math>T</math>, is given by <math>T = P^{-1}SP</math>. The transform in the original basis is found to be the product of three easy-to-derive matrices. In effect, the similarity transform operates in three steps: change to a new basis ({{mvar|P}}), perform the simple transformation ({{mvar|S}}), and change back to the old basis ({{math|''P''<sup>−1</sup>}}).