Linear Algebra/Diagonalizability/Solutions

SolutionsEdit

This exercise is recommended for all readers.
Problem 1

Repeat Example 2.5 for the matrix from Example 2.2.

Answer

Because the basis vectors are chosen arbitrarily, many different answers are possible. However, here is one way to go; to diagonalize

 

take it as the representation of a transformation with respect to the standard basis   and look for   such that

 

that is, such that   and  .

 

We are looking for scalars   such that this equation

 

has solutions   and  , which are not both zero. Rewrite that as a linear system

 

If   then the first equation gives that  , and then the second equation gives that  . The case where both  's are zero is disallowed so we can assume that  .

 

Consider the bottom equation. If   then the first equation gives   or  . The   case is disallowed. The other possibility for the bottom equation is that the numerator of the fraction   is zero. The   case gives a first equation of  , and so associated with   we have vectors whose first and second components are equal:

 

If   then the first equation is   and so the associated vectors are those whose first component is twice their second:

 

This picture

 

shows how to get the diagonalization.

 

Comment. This equation matches the   definition under this renaming.

 
Problem 2

Diagonalize these upper triangular matrices.

  1.  
  2.  
Answer
  1. Setting up
     
    gives the two possibilities that   and  . Following the   possibility leads to the first equation   with the two cases that   and that  . Thus, under this first possibility, we find   and the associated vectors whose second component is zero, and whose first component is free.
     
    Following the other possibility leads to a first equation of   and so the vectors associated with this solution have a second component that is four times their first component.
     
    The diagonalization is this.
     
  2. The calculations are like those in the prior part.
     
    The bottom equation gives the two possibilities that   and  . Following the   possibility, and discarding the case where both   and   are zero, gives that  , associated with vectors whose second component is zero and whose first component is free.
     
    The   possibility gives a first equation of   and so the associated vectors have a second component that is the negative of their first component.
     
    We thus have this diagonalization.
     
This exercise is recommended for all readers.
Problem 3

What form do the powers of a diagonal matrix have?

Answer

For any integer  ,

 
Problem 4

Give two same-sized diagonal matrices that are not similar. Must any two different diagonal matrices come from different similarity classes?

Answer

These two are not similar

 

because each is alone in its similarity class.

For the second half, these

 

are similar via the matrix that changes bases from   to  . (Question. Are two diagonal matrices similar if and only if their diagonal entries are permutations of each other's?)

Problem 5

Give a nonsingular diagonal matrix. Can a diagonal matrix ever be singular?

Answer

Contrast these two.

 

The first is nonsingular, the second is singular.

This exercise is recommended for all readers.
Problem 6

Show that the inverse of a diagonal matrix is the diagonal of the inverses, if no element on that diagonal is zero. What happens when a diagonal entry is zero?

Answer

To check that the inverse of a diagonal matrix is the diagonal matrix of the inverses, just multiply.

 

(Showing that it is a left inverse is just as easy.)

If a diagonal entry is zero then the diagonal matrix is singular; it has a zero determinant.

Problem 7

The equation ending Example 2.5

 

is a bit jarring because for   we must take the first matrix, which is shown as an inverse, and for   we take the inverse of the first matrix, so that the two   powers cancel and this matrix is shown without a superscript  .

  1. Check that this nicer-appearing equation holds.
     
  2. Is the previous item a coincidence? Or can we always switch the   and the  ?
Answer
  1. The check is easy.
     
  2. It is a coincidence, in the sense that if   then   need not equal  . Even in the case of a diagonal matrix  , the condition that   does not imply that   equals  . The matrices from Example 2.2 show this.
     
Problem 8

Show that the   used to diagonalize in Example 2.5 is not unique.

Answer

The columns of the matrix are chosen as the vectors associated with the  's. The exact choice, and the order of the choice was arbitrary. We could, for instance, get a different matrix by swapping the two columns.

Problem 9

Find a formula for the powers of this matrix Hint: see Problem 3.

 
Answer

Diagonalizing and then taking powers of the diagonal matrix shows that

 
This exercise is recommended for all readers.
Problem 10

Diagonalize these.

  1.  
  2.  
Answer
  1.  
  2.  
Problem 11

We can ask how diagonalization interacts with the matrix operations. Assume that   are each diagonalizable. Is   diagonalizable for all scalars  ? What about  ?  ?

Answer

Yes,   is diagonalizable by the final theorem of this subsection.

No,   need not be diagonalizable. Intuitively, the problem arises when the two maps diagonalize with respect to different bases (that is, when they are not simultaneously diagonalizable). Specifically, these two are diagonalizable but their sum is not:

 

(the second is already diagonal; for the first, see Problem 10). The sum is not diagonalizable because its square is the zero matrix.

The same intuition suggests that   is not be diagonalizable. These two are diagonalizable but their product is not:

 

(for the second, see Problem 10).

This exercise is recommended for all readers.
Problem 12

Show that matrices of this form are not diagonalizable.

 
Answer

If

 

then

 

so

 

The   entries show that   and the   entries then show that  . Since   this means that  . The   entries show that   and the   entries then show that  . Since   this means that  . But if both   and   are   then   is not invertible.

Problem 13

Show that each of these is diagonalizable.

  1.  
  2.  
Answer
  1. Using the formula for the inverse of a   matrix gives this.
     
    Now pick scalars   so that   and   and  . For example, these will do.
     
  2. As above,
     
    we are looking for scalars   so that   and   and  , no matter what values  ,  , and   have. For starters, we assume that  , else the given matrix is already diagonal. We shall use that assumption because if we (arbitrarily) let   then we get
     
    and the quadratic formula gives
     
    (note that if  ,  , and   are real then these two  's are real as the discriminant is positive). By the same token, if we (arbitrarily) let   then
     
    and we get here
     
    (as above, if   then this discriminant is positive so a symmetric, real,   matrix is similar to a real diagonal matrix). For a check we try  ,  ,  .
     
    Note that not all four choices   satisfy  .