- 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.
- 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 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.
- This exercise is recommended for all readers.
- Problem 13
Show that each of these is diagonalizable.
-
-
- Answer
- Using the formula for the inverse of a
matrix gives this.
Now pick scalars so that
and and .
For example, these will do.
- 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 .