Linear Algebra/Changing Map Representations/Solutions

SolutionsEdit

This exercise is recommended for all readers.
Problem 1

Decide if these matrices are matrix equivalent.

  1.  ,  
  2.  ,  
  3.  ,  
Answer
  1. Yes, each has rank two.
  2. Yes, they have the same rank.
  3. No, they have different ranks.
This exercise is recommended for all readers.
Problem 2

Find the canonical representative of the matrix-equivalence class of each matrix.

  1.  
  2.  
Answer

We need only decide what the rank of each is.

  1.  
  2.  
Problem 3

Suppose that, with respect to

 

the transformation   is represented by this matrix.

 

Use change of basis matrices to represent   with respect to each pair.

  1.  ,  
  2.  ,  
Answer

Recall the diagram and the formula.

   
  1. These two
     
    show that
     
    and similarly these two
     
    give the other nonsinguar matrix.
     
    Then the answer is this.
     
    Although not strictly necessary, a check is reassuring. Arbitrarily fixing
     
    we have that
     
    and so   is this.
     
    Doing the calculation with respect to   starts with
     
    and then checks that this is the same result.
     
  2. These two
     
    show that
     
    and these two
     
    show this.
     
    With those, the conversion goes in this way.
     
    As in the prior item, a check provides some confidence that this calculation was performed without mistakes. We can for instance, fix the vector
     
    (this is selected for no reason, out of thin air). Now we have
     
    and so   is this vector.
     
    With respect to   we first calculate
     
    and, sure enough, that is the same result for  .
     
This exercise is recommended for all readers.
Problem 4

What sizes are   and   in the equation  ?

Answer

Where   and   are  , the matrix   is   while   is  .

This exercise is recommended for all readers.
Problem 5

Use Theorem 2.6 to show that a square matrix is nonsingular if and only if it is equivalent to an identity matrix.

Answer

Any   matrix is nonsingular if and only if it has rank  , that is, by Theorem 2.6, if and only if it is matrix equivalent to the   matrix whose diagonal is all ones.

This exercise is recommended for all readers.
Problem 6

Show that, where   is a nonsingular square matrix, if   and   are nonsingular square matrices such that   then  .

Answer

If   then  , so  , and so  .

This exercise is recommended for all readers.
Problem 7

Why does Theorem 2.6 not show that every matrix is diagonalizable (see Example 2.2)?

Answer

By the definition following Example 2.2, a matrix   is diagonalizable if it represents   a transformation with the property that there is some basis   such that   is a diagonal matrix— the starting and ending bases must be equal. But Theorem 2.6 says only that there are   and   such that we can change to a representation   and get a diagonal matrix. We have no reason to suspect that we could pick the two   and   so that they are equal.

Problem 8

Must matrix equivalent matrices have matrix equivalent transposes?

Answer

Yes. Row rank equals column rank, so the rank of the transpose equals the rank of the matrix. Same-sized matrices with equal ranks are matrix equivalent.

Problem 9

What happens in Theorem 2.6 if  ?

Answer

Only a zero matrix has rank zero.

This exercise is recommended for all readers.
Problem 10

Show that matrix-equivalence is an equivalence relation.

Answer

For reflexivity, to show that any matrix is matrix equivalent to itself, take   and   to be identity matrices. For symmetry, if   then   (inverses exist because   and   are nonsingular). Finally, for transitivity, assume that   and that  . Then substitution gives  . A product of nonsingular matrices is nonsingular (we've shown that the product of invertible matrices is invertible; in fact, we've shown how to calculate the inverse) and so   is therefore matrix equivalent to  .

This exercise is recommended for all readers.
Problem 11

Show that a zero matrix is alone in its matrix equivalence class. Are there other matrices like that?

Answer

By Theorem 2.6, a zero matrix is alone in its class because it is the only   of rank zero. No other matrix is alone in its class; any nonzero scalar product of a matrix has the same rank as that matrix.

Problem 12

What are the matrix equivalence classes of matrices of transformations on  ?  ?

Answer

There are two matrix-equivalence classes of   matrices— those of rank zero and those of rank one. The   matrices fall into four matrix equivalence classes.

Problem 13

How many matrix equivalence classes are there?

Answer

For   matrices there are classes for each possible rank: where   is the minimum of   and   there are classes for the matrices of rank  ,  , ...,  . That's   classes. (Of course, totaling over all sizes of matrices we get infinitely many classes.)

Problem 14

Are matrix equivalence classes closed under scalar multiplication? Addition?

Answer

They are closed under nonzero scalar multiplication, since a nonzero scalar multiple of a matrix has the same rank as does the matrix. They are not closed under addition, for instance,   has rank zero.

Problem 15

Let   represented by   with respect to  .

  1. Find   in this specific case.
     
  2. Describe   in the general case where  .
Answer
  1. We have
     
    and thus the answer is this.
     
    As a quick check, we can take a vector at random
     
    giving
     
    while the calculation with respect to  
     
    yields the same result.
     
  2. We have
       

    and, as in the first item of this question

     

    so, writing   for the matrix whose columns are the basis vectors, we have that  .

Problem 16
  1. Let   have bases   and   and suppose that   has the basis  . Where  , find the formula that computes   from  .
  2. Repeat the prior question with one basis for   and two bases for  .
Answer
  1. The adapted form of the arrow diagram is this.
     
    Since there is no need to change bases in   (or we can say that the change of basis matrix   is the identity), we have   where  .
  2. Here, this is the arrow diagram.
     
    We have that   where  .
Problem 17
  1. If two matrices are matrix-equivalent and invertible, must their inverses be matrix-equivalent?
  2. If two matrices have matrix-equivalent inverses, must the two be matrix-equivalent?
  3. If two matrices are square and matrix-equivalent, must their squares be matrix-equivalent?
  4. If two matrices are square and have matrix-equivalent squares, must they be matrix-equivalent?
Answer
  1. Here is the arrow diagram, and a version of that diagram for inverse functions.
       

    Yes, the inverses of the matrices represent the inverses of the maps. That is, we can move from the lower right to the lower left by moving up, then left, then down. In other words, where   (and   invertible) and   are invertible then  .

  2. Yes; this is the prior part repeated in different terms.
  3. No, we need another assumption: if   represents   with respect to the same starting as ending bases  , for some   then   represents  . As a specific example, these two matrices are both rank one and so they are matrix equivalent
     
    but the squares are not matrix equivalent— the square of the first has rank one while the square of the second has rank zero.
  4. No. These two are not matrix equivalent but have matrix equivalent squares.
     
This exercise is recommended for all readers.
Problem 18

Square matrices are similar if they represent the same transformation, but each with respect to the same ending as starting basis. That is,   is similar to  .

  1. Give a definition of matrix similarity like that of Definition 2.3.
  2. Prove that similar matrices are matrix equivalent.
  3. Show that similarity is an equivalence relation.
  4. Show that if   is similar to   then   is similar to  , the cubes are similar, etc. Contrast with the prior exercise.
  5. Prove that there are matrix equivalent matrices that are not similar.
Answer
  1. The definition is suggested by the appropriate arrow diagram.
     
    Call matrices   similar if there is a nonsingular matrix   such that  .
  2. Take   to be   and take   to be  .
  3. This is as in Problem 10. Reflexivity is obvious:  . Symmetry is also easy:   implies that   (multiply the first equation from the right by   and from the left by  ). For transitivity, assume that   and that  . Then   and we are finished on noting that   is an invertible matrix with inverse  .
  4. Assume that  . For the squares:  . Higher powers follow by induction.
  5. These two are matrix equivalent but their squares are not matrix equivalent.
     
    By the prior item, matrix similarity and matrix equivalence are thus different.