University Mathematics Handbook (2015)
X. Algebra
Chapter 10. Matrix Similarity
a. Square matrices 
and 
in 
are similar if there exists invertible matrix 
in , such that 
. It is denoted 
.
b. Properties of similarity: If
1. 
2. 
.
3. 
4. 
5. 
c. Matrices and represent the same linear operator in different bases if, and only if, they are similar.
d. 
is a diagonalizable matrix if it is similar to a diagonal matrix, that is, if there exists invertible matrix and diagonal matrix 
such that 
.
e. If is similar to 
, then is similar to 
for every permutation 
.