Equivalent Matrices: Definition & A=B

[iVenky]

What is the exact definition for equivalent matrices?

Is it necessary that it should be A = B if A,B are two matrices?

Thanks a lot.

 

[HallsofIvy]

The standard definition of "equivalent" for matrices (in any Linear Algebra text) is
"Matrices A and C are equivalent if and only if there exist an invertible matrix, B, such that BA= CB." Since B is invertible, that is the same as saying that $A= B^{-1}CB$ as well as $C= BAB^{-1}$. From a more abstract point of view, matrices A and C are equivalent if and only if they represent the same linear transformation, on some vector spaces, as represented in different bases.