similarity
Two square matrices A and B are similar if B = P^-1 A P for some invertible P. The point is not the formula but its meaning: A and B are similar exactly when they are matrices of the SAME operator written in two different bases. Similarity is the equivalence relation that says do not let a change of coordinates fool you into thinking these are different objects.
Here P is the change-of-basis matrix. If A is the matrix of T in basis B and you switch to a new basis whose vectors are the columns of P, the matrix of the very same T becomes P^-1 A P. So similarity classes are in bijection with operators-up-to-basis. Two matrices are similar iff they represent one geometric action.
Because similar matrices are the same operator in disguise, every genuine property of the operator is a similarity invariant: they share determinant, trace, rank, characteristic polynomial, minimal polynomial, eigenvalues with their multiplicities, and Jordan structure. Conversely, anything tied to a particular basis — individual entries, being upper-triangular, being symmetric — is generally NOT preserved.
The grand project of operator theory is to choose, within each similarity class, the simplest possible representative — a canonical form. Over an algebraically closed field that representative is the Jordan form; over the reals one uses the real Jordan or rational canonical form. Two matrices are similar if and only if they have the same canonical form, which is exactly why canonical forms classify operators.
Conjugating by an invertible P leaves every operator invariant untouched; only the coordinate clothing changes.
Similarity is finer than equivalence. Any two matrices of the same size and rank are equivalent (B = Q^-1 A P with two independent changes), but similarity demands the SAME change on both sides, so it preserves much more — eigenvalues included. Equivalence sees only rank; similarity sees the whole operator.