Page 216 - Advanced Linear Algebra

200 Advanced Linear Algebra

14. Give an example of a complex nonreal matrix all of whose eigenvalues are
real. Show that any such matrix is similar to a real matrix. What about the
type of the invertible matrices that are used to bring the matrix to Jordan
form?

B
be the Jordan form of a linear operator  ²= ³ . For a given
15. Let 1~ ´ µ 8
Jordan block of 1² Á ³ let be the subspace of spanned by the basis
=
<

vectors of associated with that block.
8
a Show that has a single eigenvalue with geometric multiplicity .
)
O
<
In other words, there is essentially only one eigenvector up to scalar
(
multiple associated with each Jordan block. Hence, the geometric
)
multiplicity of for is the number of Jordan blocks for . Show that

the algebraic multiplicity is the sum of the dimensions of the Jordan
blocks associated with .

)
b Show that the number of Jordan blocks in is the maximum number
1
of linearly independent eigenvectors of .

)
c What can you say about the Jordan blocks if the algebraic multiplicity
of every eigenvalue is equal to its geometric multiplicity?
16. Assume that the base field is algebraically closed. Then assuming that the
-
eigenvalues of a matrix are known, it is possible to determine the Jordan
(
form of by looking at the rank of various matrix powers. A matrix is
)
(
1

nilpotent if )~ for some € . The smallest such exponent is called
the index of nilpotence .
)
a Let 1~ 1² Á ³ be a single Jordan block of size d . Show that

1c 0 is nilpotent of index . Thus, is the smallest integer for

which rk²1 c 0³ ~ .

Now let be a matrix in Jordan form but possessing only one eigenvalue
1
.
b Show that 1c 0 is nilpotent. Let be its index of nilpotence. Show
)

1
that is the maximum size of the Jordan blocks of and that

rk²1 c 0³ c is the number of Jordan blocks in 1 of maximum size.
)
c Show that rk²1 c 0³ c is equal to times the number of Jordan

blocks of maximum size plus the number of Jordan blocks of size one
less than the maximum.
d Show that the sequence rk²1 c 0³ for ~ ÁÃÁ uniquely
)

determines the number and size of all of the Jordan blocks in , that is,
1
it uniquely determines up to the order of the blocks.
1
)
e Now let be an arbitrary Jordan matrix. If is an eigenvalue for 1
1

show that the sequence rk²1 c 0³ for ~ ÁÃÁ where is the

first integer for which rk²1 c 0³ ~ rk²1 c 0³ b uniquely

determines up to the order of the blocks.
1
f Prove that for any matrix with spectrum ¸ Á Ã Á ¹ the sequence
)
(

0³ for ~ ÁÃÁ and ~ Á ÃÁ where is the first
rk²( c
integer for which rk²( c 0³ ~ rk²( c 0³ b uniquely

determines the Jordan matrix for up to the order of the blocks.
(
1
17. Let ( C ²- . ³