Similar or Conjugate Matrices

Matrices  
\[A\]
  and  
\[B\]
  are similar if there exists an invertible matrix  
\[P\]
  such that  
\[B=P^{-1}AP\]

The matrices  
\[ \left( \begin{array}{cc} 4 & 3 \\ 2 & 1 \end{array} \right) \]
  and  
\[ \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) \]
  are similar with  
\[P= \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \]

Note that  
\[ \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)^{-1} =\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \]
  so that  
\[P\]
  is self inverse.
\[ \left( \begin{array}{cc} 4 & 3 \\ 2 & 1 \end{array} \right) = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)^{-1} \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \]

\[B\]
  is also said to be conjugate to  
\[B\]
  and vice versa.
Only square matrices can be similar.

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