Page 184 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes The T-conductor of  into W

Let W be an invariant subspace for T and let  be a vector in V. The T-conductor of  into W is the
set S (; W), which consists of all polynomials g (over the scalar field), such that g(T) is in W.
T
Some authors call that collection of polynomials the ‘stuffer’. In the special case W = {0}, the
conductor is called T-annihilator of .

14.2 Theorems and Examples

Example 1: If T is any linear operator on V, then V is invariant under T, as is the zero
subspace. The range of T and the null space of T are also invariant under T.

Example 2: Let F be a field and let D be the differentiation operator on the space F[x] of
polynomials over F. Let n be a positive integer and let W be the subspace of polynomials of
degree not greater than n. Then W is invariant under D. This is just another way of saying that D
is ‘degree decreasing’.

Example 3: Here is a very useful generalization of Example 1. Let T be a linear operator
on V. Let U be any linear operator on V which commutes with T, i.e., TU = UT. Let W be the range
of U and let N be the null space of U. Both W and N are invariant under T. If  is in the range
of U, say  = U, then T = T(U) = U(T) so that T is in the range of U. If  is in N, then U(T)
= T(U) = T(0) = 0; hence T is in N.
A particular type of operator which commutes with T is an operator U = g(T), where g is a
polynomial. For instance, we might have U = T – cI, where c is a characteristic value of T. The null
space of U is familiar to us. We see that this example includes the (obvious) fact that the space of
characteristic vectors of T associated with the characteristic value c is invariant under T.

2
Example 4: Let T be the linear operator on R which is represented in the standard
ordered basis by the matrix

1
 0  
A    .
 1 0 
Then the only subspaces of R which are invariant under T are R and the zero subspace. Any
2
2
other invariant subspace would necessarily have dimension 1. But, if W is the subspace spanned
by some non-zero vector , the fact that W is invariant under T means that  is a characteristic
vector, but A has no real characteristic values.
When V is finite-dimensional, the invariance of W under T has a simple matrix interpretation,
and perhaps we should mention it at this point. Suppose we choose an ordered basis  = { ,..., }
1 n
for V such that ’ = { ,..., } is an ordered basis for W (r = dim W). Let A = [T] so that
1 r 
n
T =  A  ...(1)
j ij i
i 1
Since W is invariant under T, the vector T belongs to W for j  r. This means that
j
r
T =  A  , j  r ...(2)
j ij i
i 1

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