Devin Anderson

2022-06-29

If I take a particular number, for example 21, and write down all possible matrices whose determinant is 21, will all these matrices represent the same transformation with different system of bases? Consider all endomorphisms.

marktje28

Expert

No. if two matrices $A$ and $B$ represent the same linear transformation, then they are related by conjugation,$B={P}^{-1}AP$, where $P$ is the change of basis. Such matrices are called similar. Two matrices may have same determinant but not be similar. They are similar if and only if they have the same canonical form.
For example
$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
and
$\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$
Both have determinant 1, but they are not similar.

Do you have a similar question?