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Data Analytics: Summaries on Strang's Linear Algebra - Similar Matrices PDF Print E-mail
Written by FemiByte   
Thursday, 21 March 2013 08:29

Similar Matrices

Let $M$ be any invertible matrix. Then $B=M^{-1}AM$ is similar to $A$. Similar matrices $A$ and $M^{-1}AM$ have the same eigenvalues. If $x$ is an eigenvector of $A$, then $M^{-1}x$ is an eigenvector of $B=M^{-1}AM$. Matrices that are similar to each other form a family of related matrices, all having the same eigenvalues.

If $A$ has $s$ independent eigenvectors, it is similar to a matrix $J$ that has $s$ Jordan blocks on its diagonal: Some matrix $M$ puts $A$ into Jordan form:

Jordan form $M^{-1}AM=\left[\begin{array}{ccccc}J_1&& && && && \\ &&.&& && &&\\ && &&.&& &&\\ && && && &&J_s&&\end{array}\right] = J$
Each block in $J$ has one eigenvalue $\lambda_i$, one eigenvector and 1's above the diagonal:
Jordan block $J_i=\left[\begin{array}{ccccc}\lambda_i&&1&& && && \\ &&. &&. && &&\\ && &&.&& &&\\ && && && &&\lambda_i&&\end{array}\right] = J$
$A$ is similar to $B$ if they share the same Jordan form $J$ - not otherwise.

Last Updated on Thursday, 21 March 2013 08:30
 

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