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The column space or range of a matrix is defined to be the span of its column vectors. If
then The rank of B is the dimension of the range: Then the geometric multiplicity of is . The null space or kernel of a matrix B is the set of vectors that the matrix send to zero:
.
For example, consider an eigenvalue and eigenvector of matrix . These satisfy
or equivalently
. Then
.
The dimension of the kernel is called the nullity of a matrix:
. 8
If matrix has columns, a theorem of linear algebra implies that
. Continue the example. If the geometric multiplicity of is
eigenvectors associated with and
. then has linearly independent Return to the analysis of the boundary states in the plane on the parabola
where is the trace of and is the determinant of . Then [6] implies that the matrix
an eigenvalue of algebraic multiplicity 2. ,
has When the geometric multiplicity of is 2,
. That is,
for
every vector
. This implies
. Every nonzero vector is an eigenvector and the
solution of
is uniform expansion away from the origin, for
, or uniform contraction
toward the origin, for
.
When the geometric multiplicity of is 1, the matrix has only a single eigenvector. This
means [4] cannot be the general solution of the initial value problem. The solution will be
obtained in section 2.6. 2.3 Exponentials of Operators
Let be a vector space. We may have
or , but results in this section also apply to
infinite dimensional vector spaces. An operator
maps a vector
to some other
vector
. A linear operator T satisfies linear superposition and scaling. If
and
and
are scalars then
.
If
and
is a basis for then T will be represented by a matrix in that basis.
In this case may replaced by in the equations below. We write
, the space of
linear operators on
is itself an
dimensional vector space. Notice that a matrix
representing has
components.
If
then
operator norm represents the Euclidean norm: . We will also define an The operator norm satisfies the usual properties for norms. In particular, the triangle inequality
is satisfied. If S and T are operators on then
. [The matrix norm is a continuous function of its argument. See Hirsch and Smale p. 78.] 9
A linear operator also satisfies since, for
. For any integer , we have by the same logic.
, Since this is true for all
[6] we conclude . An operator for which is said to be bounded. Example. Suppose
. On a finite dimensional vector space the sphere
is
comp...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
 Fall '14
 MarcEvans

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