1: Direct sums and Jordan forms
In this chapter we take as a central theme the notion of the direct sum
A 0 B of subspaces A, B of a vector space V. Recall that V = A 0 B
if and only if every x G V can be expressed uniquely in the form a + b
where aeA and b e B; equivalently, if V = A+B and AnB = {0}. For
every subspace A of V there is a subspace B of V such that V = A®B.
In the case where V is of finite dimension, this is easily seen; take a basis
{vi,..., Vk} of A, extend it to a basis {^i,..., vn} of V, then note that
{vk+i,..., vn} spans a subspace 5 such that V = A 0 B.
If / : V —> V is a linear transformation then a subspace W of V is
said to be {^invariant if / maps W into itself. If W is /-invariant then
there is an ordered basis of V with respect to which the matrix of V is
of the form
M N
0
where M is of size dim W x dimW.
If / : V —> V is such that / o / = / then / is called a projection.
For such a linear transformation we have V = Im / 0 Ker / where the
subspace I m / is /-invariant (and the subspace Ker/ is trivially so). A
vector space V is the direct sum of subspaces Wx,..., Wk if and only if
there are non-zero projections P i , . . . ,p* : V —> V such that
Pi — pi o pj = 0 for t ^ ;.
4=1
In this case W{ = Imp,- for each i, and relative to given ordered bases of
,Book 4 Linear algebra
Wi,...,Wk the matrix of / is of the diagonal block form
M2
Mk
Of particular importance is the situation where each M,- is of the form
A 1 0 ... 0 0
0 A 1 ... 0 0
0 0 A ... 0 0
M,=
. . .
0 0 0 ... A 1
0 0 0 ... 0 A
in which case the diagonal block matrix is called a Jordan matrix.
The Cayley-Hamilton theorem says that a linear transformation / is
a zero of its characteristic polynomial. The minimum polynomial of /
is the monic polynomial of least degree of which / is a zero. When the
minimum polynomial of / factorises into a product of linear polynomials
then there is a basis of V with respect to which the matrix of / is a
Jordan matrix. This matrix is unique (up to the sequence of the diagonal
blocks), the diagonal entries A above are the eigenvalues of / , and the
number of Mt- associated with a given A is the geometric multiplicity of
A. The corresponding basis is called a Jordan basis.
We mention here that, for space considerations in the solutions, we
shall often write an eigenvector
as i, x2,
1.1 Which of the following statements are true? For those that are false,
give a counter-example.
(i) If {0,1,(12, a$} is a basis for IR3 and b is a non-zero vector in IR then
{b + ai, ci2, as} is also a basis for IR3.
, 1: Direct sums and Jordan forms
(ii) If A is a finite set of linearly independent vectors then the dimension
of the subspace spanned by A is equal to the number of vectors in
A.
(iii) The subspace {(z,x, x) \ x £ IR} of IR3 has dimension 3.
(iv) If A is a linearly dependent set of vectors in IRn then there are more
than n vectors in A.
(v) If A is a linearly dependent subset of IRn then the dimension of the
subspace spanned by A is strictly less than the number of vectors
in A.
(vi) If A is a subset of IRn and the subspace spanned by A is \Rn itself
then A contains exactly n vectors,
(vii) If A and B are subspaces of IRn then we can find a basis of IRn
which contains a basis of A and a basis of B.
(viii) An n-dimensional vector space contains only finitely many sub-
spaces.
(ix) If A is an n x n matrix over Q with A3 = I then A is non-singular,
(x) If A is an n x n matrix over C with A3 = I then A is non-singular,
(xi) An isomorphism between two vector spaces can always be repre-
sented by a square singular matrix.
(xii) Any two n-dimensional vector spaces are isomorphic.
(xiii) If A is an n x n matrix such that A2 = I then A — I.
(xiv) If A, B and C are non-zero matrices such that AC = BC then
A = B.
(xv) The identity map on IRn is represented by the identity matrix with
respect to any basis of IRn.
(xvi) Given any two bases of IRn there is an isomorphism from IRn to
itself that maps one basis onto the other.
(xvii) If A and B represent linear transformations /, g : IRn —> IR™ with
respect to the same basis then there is a non-singular matrix P such
that P~lAP = B.
(xviii) There is a bijection between the set of linear transformations from
IRn to itself and the set of n x n matrices over IR.
(xix) The map t : IR2 -* IR2 given by t(x, y) — (y, x+y) can be represented
by the matrix
r
1 2
1 2
with respect to some basis of IR2.
(xx) There is a non-singular matrix P such that P~XAP is diagonal for
any non-singular matrix A.
In this chapter we take as a central theme the notion of the direct sum
A 0 B of subspaces A, B of a vector space V. Recall that V = A 0 B
if and only if every x G V can be expressed uniquely in the form a + b
where aeA and b e B; equivalently, if V = A+B and AnB = {0}. For
every subspace A of V there is a subspace B of V such that V = A®B.
In the case where V is of finite dimension, this is easily seen; take a basis
{vi,..., Vk} of A, extend it to a basis {^i,..., vn} of V, then note that
{vk+i,..., vn} spans a subspace 5 such that V = A 0 B.
If / : V —> V is a linear transformation then a subspace W of V is
said to be {^invariant if / maps W into itself. If W is /-invariant then
there is an ordered basis of V with respect to which the matrix of V is
of the form
M N
0
where M is of size dim W x dimW.
If / : V —> V is such that / o / = / then / is called a projection.
For such a linear transformation we have V = Im / 0 Ker / where the
subspace I m / is /-invariant (and the subspace Ker/ is trivially so). A
vector space V is the direct sum of subspaces Wx,..., Wk if and only if
there are non-zero projections P i , . . . ,p* : V —> V such that
Pi — pi o pj = 0 for t ^ ;.
4=1
In this case W{ = Imp,- for each i, and relative to given ordered bases of
,Book 4 Linear algebra
Wi,...,Wk the matrix of / is of the diagonal block form
M2
Mk
Of particular importance is the situation where each M,- is of the form
A 1 0 ... 0 0
0 A 1 ... 0 0
0 0 A ... 0 0
M,=
. . .
0 0 0 ... A 1
0 0 0 ... 0 A
in which case the diagonal block matrix is called a Jordan matrix.
The Cayley-Hamilton theorem says that a linear transformation / is
a zero of its characteristic polynomial. The minimum polynomial of /
is the monic polynomial of least degree of which / is a zero. When the
minimum polynomial of / factorises into a product of linear polynomials
then there is a basis of V with respect to which the matrix of / is a
Jordan matrix. This matrix is unique (up to the sequence of the diagonal
blocks), the diagonal entries A above are the eigenvalues of / , and the
number of Mt- associated with a given A is the geometric multiplicity of
A. The corresponding basis is called a Jordan basis.
We mention here that, for space considerations in the solutions, we
shall often write an eigenvector
as i, x2,
1.1 Which of the following statements are true? For those that are false,
give a counter-example.
(i) If {0,1,(12, a$} is a basis for IR3 and b is a non-zero vector in IR then
{b + ai, ci2, as} is also a basis for IR3.
, 1: Direct sums and Jordan forms
(ii) If A is a finite set of linearly independent vectors then the dimension
of the subspace spanned by A is equal to the number of vectors in
A.
(iii) The subspace {(z,x, x) \ x £ IR} of IR3 has dimension 3.
(iv) If A is a linearly dependent set of vectors in IRn then there are more
than n vectors in A.
(v) If A is a linearly dependent subset of IRn then the dimension of the
subspace spanned by A is strictly less than the number of vectors
in A.
(vi) If A is a subset of IRn and the subspace spanned by A is \Rn itself
then A contains exactly n vectors,
(vii) If A and B are subspaces of IRn then we can find a basis of IRn
which contains a basis of A and a basis of B.
(viii) An n-dimensional vector space contains only finitely many sub-
spaces.
(ix) If A is an n x n matrix over Q with A3 = I then A is non-singular,
(x) If A is an n x n matrix over C with A3 = I then A is non-singular,
(xi) An isomorphism between two vector spaces can always be repre-
sented by a square singular matrix.
(xii) Any two n-dimensional vector spaces are isomorphic.
(xiii) If A is an n x n matrix such that A2 = I then A — I.
(xiv) If A, B and C are non-zero matrices such that AC = BC then
A = B.
(xv) The identity map on IRn is represented by the identity matrix with
respect to any basis of IRn.
(xvi) Given any two bases of IRn there is an isomorphism from IRn to
itself that maps one basis onto the other.
(xvii) If A and B represent linear transformations /, g : IRn —> IR™ with
respect to the same basis then there is a non-singular matrix P such
that P~lAP = B.
(xviii) There is a bijection between the set of linear transformations from
IRn to itself and the set of n x n matrices over IR.
(xix) The map t : IR2 -* IR2 given by t(x, y) — (y, x+y) can be represented
by the matrix
r
1 2
1 2
with respect to some basis of IR2.
(xx) There is a non-singular matrix P such that P~XAP is diagonal for
any non-singular matrix A.
