Linear Algebra/Definition and Examples of Similarity/Solutions

SolutionsEdit

Problem 1

For

 

check that  .

Answer

One way to proceed is left to right.

 
This exercise is recommended for all readers.
Problem 2

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.

  1. Show that the   matrix  , also, is similar only to itself.
  2. Is a matrix of the form   for some scalar   similar only to itself?
  3. Is a diagonal matrix similar only to itself?
Answer
  1. Because the matrix   is  , the matrices   and   are also   and so where   the inverse is  . Thus  .
  2. 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  .
  3. 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  .

  1. Find   where  .
  2. Find   where  .
  3. Find the matrix   such that  .
Answer
  1. Because   is described with the members of  , finding the matrix representation is easy:
     
    gives this.
     
  2. 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.
     
  3. 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.

 

Therefore

 

and these are the change of basis matrices.

 

The check of these computations is routine.

 
Problem 6

Explain Example 1.3 in terms of maps.

Answer

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  .)

ReferencesEdit

  • Halmos, Paul P. (1958), Finite Dimensional Vector Spaces (Second ed.), Van Nostrand .