This subsection moves from the canonical form for nilpotent matrices to
the one for all matrices.
We have shown that if a map is nilpotent then all of its eigenvalues are zero.
We can now prove the converse.
- Lemma 2.1
A linear transformation whose only eigenvalue is zero is nilpotent.
If a transformation on an
-dimensional space has only the single eigenvalue of zero
then its characteristic polynomial is .
The Cayley-Hamilton Theorem says that a map satisfies its
characteristic polynimial so is the zero map.
Thus is nilpotent.
We have a canonical form for nilpotent matrices,
that is, for each matrix whose single eigenvalue is zero: each
such matrix is similar to one that is all
zeroes except for blocks of subdiagonal ones.
(To make this representation unique we can fix some arrangement of
the blocks, say, from longest to shortest.)
We next extend this to all single-eigenvalue matrices.
Observe that if 's only eigenvalue is then
's only eigenvalue is because
only if .
The natural way to extend the results for nilpotent matrices is to
represent in the canonical form , and try to use that
to get a simple representation for .
The next result says that this try works.
- Lemma 2.2
If the matrices and are similar
then and are also similar,
via the same change of basis matrices.
since the diagonal matrix commutes with anything,
and so .
Therefore , as required.
- Example 2.3
The characteristic polynomial of
is and so has only the single eigenvalue .
the only eigenvalue is , and is nilpotent.
The null spaces are routine to find; to ease this computation we take
to represent the transformation with respect to
the standard basis (we shall maintain this convention
for the rest of the chapter).
The dimensions of these null spaces
show that the action of an associated map on a string basis is
Thus, the canonical form for
with one choice for a string basis is
and by Lemma 2.2, is similar to
We can produce the similarity computation.
Recall from the Nilpotence section how to find the change of
basis matrices and to express as .
The similarity diagram
describes that to move from the lower left to the upper left we multiply by
and to move from the upper right to the lower right we multiply by
So the similarity is expressed by
which is easily checked.
- Example 2.4
This matrix has characteristic polynomial
and so has the single eigenvalue .
The nullities of are:
the null space of has dimension two, the null space of
has dimension three, and the null space of has dimension four.
Thus, has the action on a string basis of
This gives the canonical form for , which in turn gives the
form for .
An array that is all zeroes, except for some number
down the diagonal and blocks of subdiagonal ones, is a
We have shown that Jordan block matrices are
canonical representatives of the similarity classes of single-eigenvalue
- Example 2.5
The matrices whose only eigenvalue is separate into
three similarity classes.
The three classes have these canonical representatives.
In particular, this matrix
belongs to the similarity class represented by the middle one, because we have
adopted the convention of ordering the blocks of subdiagonal ones from the
longest block to the shortest.
We will now finish the program of this chapter by extending this work to
cover maps and matrices with multiple eigenvalues.
The best possibility for general maps and matrices would be
if we could break them into a part involving
their first eigenvalue
(which we represent using its Jordan block),
a part with , etc.
This ideal is in fact what happens.
For any transformation ,
we shall break the space into the direct sum of a part on which
is nilpotent, plus a part on which
is nilpotent, etc.
More precisely, we shall take three steps to get to this section's major
theorem and the third step shows that
where are 's eigenvalues.
Suppose that is a linear transformation.
Note that the restriction
of to a subspace need not be a linear transformation on
because there may be an
To ensure that the restriction of a transformation
to a "part" of a space is a transformation on the partwe need the next
- Definition 2.6
Let be a transformation.
A subspace is invariant
if whenever then
Two examples are that
the generalized null space and the generalized range space
of any transformation are invariant.
For the generalized null space, if then
where is the dimension of the underlying space
and so because
is zero also.
For the generalized range space, if then
for some and then
shows that is also a member of .
Thus the spaces and
Observe also that is nilpotent on
if has the property that
some power of maps it to zero— that is, if it is in the
generalized null space— then some power of maps
it to zero.
The generalized null space is a "part" of
the space on which the action of is easy to understand.
The next result is the first of our three steps.
It establishes that leaves
's part unchanged.
- Lemma 2.7
A subspace is invariant if and only if
it is invariant for any scalar .
where is an eigenvalue of a linear transformation
, then for any other eigenvalue ,
For the first sentence we check the two implications of the
"if and only if" separately.
One of them is easy: if the subspace is invariant for
any then taking shows that it is invariant.
For the other implication suppose that the subspace is invariant,
so that if then , and let be
The subspace is closed under linear combinations and so if
Thus if then , as required.
The second sentence follows straight from the first.
two spaces are invariant, they are
From this, applying the first sentence again, we
conclude that they are also invariant.
The second step of the three that we will take to prove this
section's major result makes use of an additional property of
, that they are complementary.
Recall that if a space is the direct sum of two others
then any vector in the space breaks into
two parts where and
, and recall also
that if and are bases for
and then the concatenation
is linearly independent (and
so the two parts of do not "overlap").
The next result says that for any subspaces
and that are complementary
as well as invariant,
of on breaks into the "non-overlapping" actions of
on and on .
- Lemma 2.8
Let be a transformation and let and
invariant complementary subspaces of .
Then can be represented by a matrix with
blocks of square submatrices and
where and are blocks of zeroes.
Since the two subspaces are complementary, the concatenation of a basis
for and a basis for makes a basis
We shall show that the matrix
has the desired form.
Any vector is in
if and only if its final
components are zeroes when it is represented with respect to .
As is invariant, each of the vectors
..., has that form.
Hence the lower left of is all zeroes.
The argument for the upper right is similar.
To see that has been decomposed into its action on the parts, observe
that the restrictions of to the subspaces
with respect to the obvious bases,
by the matrices and .
So, with subspaces that are invariant and complementary,
we can split the problem of examining
a linear transformation into two lower-dimensional subproblems.
The next result illustrates this decomposition into blocks.
- Lemma 2.9
If is a matrices with square submatrices and
where the 's are blocks of zeroes,
Suppose that is ,
that is ,
and that is .
In the permutation formula for the determinant
each term comes from a rearrangement of the column numbers
into a new order .
The upper right block is all zeroes, so if a
has at least one of among its first
column numbers then the term arising
from is zero,
e.g., if then
So the above formula reduces to a sum over all permutations with
two halves: any significant is the composition of a that
and a that rearranges only .
Now, the distributive law
(and the fact that the signum of a composition is the product
of the signums) gives that this
- Example 2.10
From Lemma 2.9 we conclude that
if two subspaces
are complementary and invariant then
is nonsingular if and only if its
to both subspaces are nonsingular.
Now for the promised third, final, step to the main result.
- Lemma 2.11
If a linear transformation has the
Because is the degree of the
characteristic polynomial, to establish statement (1) we need only show that
statement (2) holds and that
is trivial whenever .
For the latter, by Lemma 2.7,
Notice that an intersection of invariant subspaces is
invariant and so the restriction of to
is a linear transformation.
But both and are nilpotent on this subspace
and so if has any eigenvalues on the intersection
then its "only" eigenvalue is both
That cannot be, so this restriction has no eigenvalues:
(Lemma V.II.3.10 shows that
the only transformation without any eigenvalues is on the trivial space).
To prove statement (2), fix the index .
and apply Lemma 2.8.
By Lemma 2.9,
By the uniqueness clause of the Fundamental Theorem of Arithmetic,
the determinants of the blocks have the same factors as the
and the sum of the powers of these factors is the power of the factor
in the characteristic polynomial:
, ..., .
Statement (2) will be proved if we will show that and that
for all , because then the degree of
the polynomial — which equals the dimension of the
generalized null space— is as required.
For that, first,
as the restriction of to
is nilpotent on that space,
the only eigenvalue of on it is .
Thus the characteristic equation of on
And thus for all .
Now consider the restriction of to .
By Note V.III.2.2, the map
is nonsingular on
and so is not an
eigenvalue of on that subspace.
Therefore, is not a factor of ,
and so .
Our major result just
translates those steps into matrix terms.
- Theorem 2.12
Any square matrix is similar to one in Jordan form
where each is the Jordan block associated with the
eigenvalue of the original matrix (that is, is all zeroes except for
's down the diagonal and some subdiagonal ones).
Given an matrix , consider the linear map
that it represents
with respect to the standard bases.
Use the prior lemma to write
where are the eigenvalues of .
Because each is invariant,
Lemma 2.8 and the prior lemma show
that is represented by a matrix that is all zeroes except for square
blocks along the diagonal.
To make those blocks into Jordan blocks, pick each
to be a string basis for the action of on
Jordan form is a canonical form for similarity classes of square
matrices, provided that we make it unique by arranging the
Jordan blocks from least eigenvalue to greatest and then
arranging the subdiagonal blocks inside each Jordan block from
longest to shortest.
- Example 2.13
has the characteristic polynomial .
We will handle the eigenvalues and separately.
Computation of the powers, and the null spaces and nullities,
of is routine.
(Recall from Example 2.3 the convention
of taking to represent a transformation, here ,
with respect to the standard basis.)
So the generalized null space has dimension two.
We've noted that the restriction of is nilpotent on this subspace.
From the way that the nullities grow we know that the action
of on a string basis
Thus the restriction can be represented in the canonical form
where many choices of basis are possible.
Consequently, the action of the restriction of to
is represented by this matrix.
The second eigenvalue's computations are easier.
Because the power of in the characteristic polynomial is one,
the restriction of to
must be nilpotent of index one.
Its action on a string basis must be and
since it is the zero map, its canonical form
is the zero matrix.
Consequently, the canonical form for the action of on
is the matrix with the single entry .
For the basis we can use any nonzero vector from the generalized null space.
Taken together, these two give that
the Jordan form of is
where is the concatenation of and .
- Example 2.14
Contrast the prior example with
which has the same characteristic polynomial .
While the characteristic polynomial is the same,
here the action of is stable after only one application— the
of to is nilpotent of index only one.
(So the contrast with the prior example is that while
the characteristic polynomial tells us to look at the
action of the on its generalized null space, the characteristic
polynomial does not describe completely its action and we
must do some computations to find, in this example, that
the minimal polynomial is .)
The restriction of to the generalized null space acts on a string
basis as and ,
and we get this Jordan block associated with the eigenvalue .
For the other eigenvalue, the arguments for the second eigenvalue of
the prior example apply again.
The restriction of to is nilpotent of
index one (it can't be of index less than one, and since is a
factor of the characteristic polynomial to the power one it can't
be of index more than one either).
Thus 's canonical form is the zero matrix,
and the associated Jordan block is the matrix with entry .
Therefore, is diagonalizable.
(Checking that the third vector in is in the nullspace of is
- Example 2.15
A bit of computing with
shows that its characteristic polynomial
shows that the restriction of to acts on a
string basis via the two strings
A similar calculation for the other eigenvalue
shows that the restriction of to its generalized null space
acts on a string basis via the two separate strings
is similar to this Jordan form matrix.
We close with the statement that the subjects considered earlier in this
Chpater are indeed, in this sense, exhaustive.
- Corollary 2.16
Every square matrix is similar to the sum of a diagonal matrix and a nilpotent