Let’s suppose we have a linear transformation on which performs the following:

Now, the matrix representation of this transformation with respect to the standard basis is clearly

But suppose we were to use a different basis for , like say . We see that our transformation maps these basis vectors as follows:

Notice that with the vectors we have on both the left and the right above are the coordinates with respect to the standard basis. We’d like to see what the matrix representing looks like with respect to the basis, so let’s convert the vectors on the right to -coordinates. Recalling that a change of basis is simply a system of equations where the columns of the coefficient matrix are the coordinates of the basis vectors (and the inverse of this matrix if we want to go the other way), we have that

So with respect to our basis, the representation of is

We will denote this matrix as where the right-most subscript means that inputs are in -coordinates, and the left-most subscript means the outputs are in -coordinates as well. Note that we could calculate by going from coordinates to standard coordinates, using the earlier matrix, then going back to -coordinates. That is,

where refers to the -to-standard basis change of basis matrix. For notational convenience, define the following

Then the above becomes

Any matrices and for which there exists a with satisfying this equation are called *similar matrices*. Note that two matrices are similar if and only if they represent the same linear transformation, but with respect to different bases. Also note that every non-singular matrix represents a change of basis matrix. Similarity forms an equivalence relation on the set of square matrices / the set of linear transformations from a finite dimensional vector space to itself.

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