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).

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