We define what it means for two matrices to be similar. Namely, A is similar to B if there is an invertible matrix P such that PA = BP. This is slightly different than the notion of equivalent matrices (in this case there are two invertible matrices P and Q such that PA = BQ). We prove that similarity is an equivalence relationship. We also show that if two matrices are similar, then they have the same characteristic polynomial and hence the same eigenvalues. Since multiplying by an invertible matrix does not distort the rank, two similar matrices also have the same rank.
#mikethemathematician, #mikedabkowski, #profdabkowski
