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Similar matrices.

Two square matrices A,B of the same order n are similar if there exists an invertible matrix P of order n such that B= P-1AP.

Example 11.5.1  
1.
The identity matrix I is similar only to itself.
2.
The matrices $\begin{pmatrix}1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix}5 & -1 \\ -2 & 0 \end{pmatrix}$ are similar. (Check it with $P= \begin{pmatrix}1 & 1 \\ 1 & -1 \end{pmatrix}$).

Proposition 11.5.2  
1.
If A and B are similar, then |A|=|B|.
2.
If A and B are similar, then $ \Tr A = \Tr B$.
3.
Similar matrices have the same characteristic polynomial, thus the same eigenvalues (with the same multiplicities).



Noah Dana-Picard
2001-02-26
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