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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Similar Matrices">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Fri Mar 28 16:00:50 1997}
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\section{Similar Matrices\label{similar matrices}}

\begin{definition}
Two $n\times n$ matrices $A$ and $B$ are similar if 
\[
B=M^{-1}AM
\]
\end{definition}

for some invertible matrix $M$.

\begin{itemize}
\item  If $B=M^{-1}AM$, then $A$ and $B$ have the same eigenvalues. An
eigenvector $x$ of $A$ corresponds to an eigenvector $M^{-1}x$ of $B$.

\item  Similar matrices represent the same \hyperref{linear transformation}{%
}{}{MA02_06.tex#linear transformation} with respect to different bases.
\end{itemize}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Eigenvalues and eigenvectors}{}{}{%
../../Help/DM8-6.tex#Eigenvectors}}

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