We've defined and to be
matrix-equivalent if there are nonsingular matrices and
such that .
That definition is motivated by this diagram
showing that and both represent but with respect to different
pairs of bases.
We now specialize that setup to the case
where the codomain equals the domain, and
where the codomain's basis equals the domain's basis.
To move from the lower left to the lower right we can either go straight
over, or up, over, and then down.
In matrix terms,
(recall that a representation of composition like this one reads right to
The matrices and are similar if there is a nonsingular such that .
Since nonsingular matrices are square,
the similar matrices and must
be square and of the same size.
With these two,
calculation gives that is similar to
The only matrix similar to the zero matrix is itself: . The only matrix similar to the identity matrix is itself: .
Since matrix similarity is a special case of matrix equivalence,
if two matrices are similar then they are equivalent.
What about the converse: must matrix equivalent square matrices be similar?
The answer is no.
The prior example shows that the similarity classes are different
from the matrix equivalence classes, because the matrix equivalence class
of the identity consists of all nonsingular matrices of that size.
Thus, for instance, these two are
matrix equivalent but not similar.
So some matrix equivalence classes
split into two or more similarity classes— similarity gives a finer
partition than does equivalence.
This picture shows some matrix equivalence classes subdivided into
To understand the similarity relation we shall study the similarity classes.
We approach this question in the same way that we've studied both the
row equivalence and matrix equivalence relations, by finding
a canonical form for
of the similarity classes, called Jordan form.
With this canonical form, we can decide if two matrices are similar by checking
whether they reduce to the same representative.
We've also seen with both row equivalence and matrix equivalence that a
canonical form gives us insight into the ways in which members of
the same class are alike
(e.g., two identically-sized matrices are matrix equivalent
if and only if they have the same rank).