Example 1.3 shows that the only matrix similar
to a zero matrix is itself and that
the only matrix similar to the identity
is itself.
Show that the matrix ,
also, is similar only to itself.
Is a matrix of the form for some scalar
similar only to itself?
Is a diagonal matrix similar only to itself?
Answer
Because the matrix is , the matrices
and are also and so where
the inverse is .
Thus .
Yes: recall that scalar multiples can be brought out
of a matrix .
By the way, the zero and identity matrices are the special cases
and .
No, as this example shows.
Problem 3
Show that these matrices are not similar.
Answer
Gauss' method shows that the first matrix represents maps of rank two while the second matrix represents maps of rank three.
Problem 4
Consider the transformation
described by
, , and .
Find where .
Find where .
Find the matrix such that .
Answer
Because is described with the members of ,
finding the matrix representation is easy:
gives this.
We will find , , and ,
to find how each is represented with respect to .
We are given that , and the other two are easy to see:
and .
By eye, we get the representation of each vector
and thus the representation of the map.
The diagram, adapted for this and ,
shows that .
This exercise is recommended for all readers.
Problem 5
Exhibit an nontrivial similarity relationship in this way: let
act by
and pick two bases,
and represent with respect to then
and .
Then compute
the and to change bases from to and
back again.
Answer
One possible choice of the bases is
(this is suggested by the map description).
To find the matrix , solve the relations
to get , , and
.
Finding involves a bit more computation.
We first find .
The relation
gives and , and so
making
and hence acts on the first basis vector in this way.
The computation for is similar.
The relation
gives and , so
making
and hence acts on the second basis vector in this way.
The only representation of a zero map is a zero matrix, no matter what the pair of bases , and so in particular for any single basis we have . The case of the identity is related, but slightly different: the only representation of the identity map, with respect to any , is the identity . (Remark: of course, we have seen examples where and — in fact, we have seen that any nonsingular matrix is a representation of the identity map with respect to some .)
This exercise is recommended for all readers.
Problem 7
Are there two matrices and that are
similar while and are not similar?
(Halmos 1958)
Answer
No. If then .
This exercise is recommended for all readers.
Problem 8
Prove that if two matrices are similar and one is invertible then
so is the other.
Answer
Matrix similarity is a special case of matrix equivalence (if matrices are similar then they are matrix equivalent) and matrix equivalence preserves nonsingularity.
(This is an extension of the rule that similar matrices have equal determinants, which can be used as indicator if it's invertible.)
This exercise is recommended for all readers.
Problem 9
Show that similarity is an equivalence relation.
Answer
A matrix is similar to itself; take to be the identity
matrix: .
If is similar to then
and so .
Rewrite this as to conclude that
is similar to .
If is similar to and is similar to then and . Then , showing that is similar to .
Problem 10
Consider a
matrix representing, with respect to some ,
reflection across the -axis in .
Consider also
a matrix representing, with respect to some ,
reflection across the -axis.
Must they be similar?
Answer
Let and be the reflection maps (sometimes called "flip"s).
For any bases
and , the matrices and
are similar.
First note that
are similar because the second matrix is the representation of
with respect to the basis :
where .
Now the conclusion follows from the transitivity part of
Problem 9.
To finish without relying on that exercise, write and . Using the equation in the first paragraph, the first of these two becomes and rewriting the second of these two as and substituting gives the desired relationship
Thus the matrices and are similar.
Problem 11
Prove that similarity preserves determinants and rank.
Does the converse hold?
Answer
We must show that if two matrices are similar then they have the same
determinant and the same rank.
Both determinant and rank are properties of matrices that we
have already shown to be preserved by matrix equivalence.
They are therefore preserved by similarity (which is a
special case of matrix equivalence: if two matrices
are similar then they are matrix equivalent).
To prove the statement without quoting the results about
matrix equivalence, note first that
rank is a property of the map (it is the dimension of the rangespace)
and since we've shown that
the rank of a map is the rank of a representation,
it must be the same for all representations.
As for determinants,
.
The converse of the statement does not hold; for instance, there are matrices with the same determinant that are not similar. To check this, consider a nonzero matrix with a determinant of zero. It is not similar to the zero matrix, the zero matrix is similar only to itself, but they have they same determinant. The argument for rank is much the same.
Problem 12
Is there a matrix equivalence class with only one matrix similarity
class inside?
One with infinitely many similarity classes?
Answer
The matrix equivalence class containing all rank
zero matrices contains only a single matrix, the zero matrix.
Therefore it has as a subset only one similarity class.
In contrast, the matrix equivalence class of matrices of rank one consists of those matrices where . For any basis , the representation of multiplication by the scalar is , so each such matrix is alone in its similarity class. So this is a case where a matrix equivalence class splits into infinitely many similarity classes.
Problem 13
Can two different diagonal matrices be in the same similarity class?
Answer
Yes, these are similar
since, where the first matrix is for , the second matrix is for .
This exercise is recommended for all readers.
Problem 14
Prove that if two matrices are similar then their -th powers
are similar when .
What if ?
Answer
The -th powers are similar because, where each matrix represents
the map , the -th powers represent
, the composition of -many 's.
(For instance, if then .)
Restated more computationally, if then
.
Induction extends that to all powers.
For the case, suppose that is invertible and that . Note that is invertible: , and that same equation shows that is similar to . Other negative powers are now given by the first paragraph.
This exercise is recommended for all readers.
Problem 15
Let be the polynomial .
Show that if is similar to then
is similar to
.
Answer
In conceptual terms, both represent for some
transformation .
In computational terms, we have this.
Problem 16
List all of the matrix equivalence classes of matrices.
Also list the similarity classes, and describe which similarity classes are
contained inside of each matrix equivalence class.
Answer
There are two equivalence classes, (i) the class of rank zero matrices,
of which there is one:
,
and (2) the class of rank one matrices,
of which there are infinitely many:
.
Each matrix is alone in its similarity class.
That's because any transformation of a one-dimensional space
is multiplication by a scalar given by
.
Thus, for any basis ,
the matrix representing a transformation
with respect to is
.
So, contained in the matrix equivalence class is (obviously) the single similarity class consisting of the matrix . And, contained in the matrix equivalence class are the infinitely many, one-member-each, similarity classes consisting of for .
Problem 17
Does similarity preserve sums?
Answer
No.
Here is an example that has two pairs, each of two similar matrices:
and
(this example is mostly arbitrary, but not entirely, because
the center matrices on the two left sides add to the zero matrix).
Note that the sums of these similar matrices are not similar
since the zero matrix is similar only to itself.
Problem 18
Show that if and are similar matrices then
and are also similar.
Answer
If then . The diagonal matrix commutes with anything, so . Thus and consequently . (So not only are they similar, in fact they are similar via the same .)