Math. H110
NORMlite
November 14, 1999 5:50 pm
Prof. W. Kahan
Page 1/21
Notes on
Vector
and
Matrix Norms
These notes survey most important properties of norms for vectors and for linear maps from one
vector space to another,
and of maps norms induce between a vector space and its dual space.
Dual Spaces
and
Transposes
of
Vectors
Along with any space of real vectors
x
comes its dual space of linear functionals
w
T
.
The
representation of
x
by a column vector
x ,
determined by a coordinate system or
Basis
,
is
accompanied by a corresponding way to represent functionals
w
T
by row vectors
w
T
so that
always w
T
x =
w
T
x
.
A change of coordinate system will change the representations of
x
and
w
T
from
x
and
w
T
to
x
= C
–1
x
and
w
T
= w
T
C
for some suitable nonsingular matrix
C ,
keeping
w
T
x
=
w
T
x
.
But between vectors
x
and
functionals
w
T
no relationship analogous
to the relationship between a column
x
and the row
x
T
that is its transpose necessarily exists.
Relationships can be invented;
so can any arbitrary maps between one vector space and another.
For example,
given a coordinate system,
we can define a functional
x
T
for every vector
x
by
choosing arbitrarily a nonsingular matrix
T
and letting
x
T
be the functional represented by the
row
(Tx)
T
in the given coordinate system.
This defines a linear map
x
T
=
T
(
x
)
from the space
of vectors
x
to its dual space;
but whatever change of coordinates replaces column vector
x
by
x
= C
–1
x
must replace
(Tx)
T
by
(
Tx
)
T
= (Tx)
T
C = (TC
x
)
T
C
to get the same functional
x
T
.
The last equations can hold for all
x
only if
T
= C
T
TC .
In other words,
the linear map
T
(
x
)
defined by the matrix
T
in one coordinate system must be defined by
T
= C
T
TC
in the
other.
This relationship between
T
and
T
is called
Congruence
( Sylvester’s
word for it ).
Evidently matrices congruent to the same matrix are congruent to each other;
can all matrices
congruent to a given matrix
T
be recognized?
Only if
T = T
T
is real and symmetric does this
question have a simple answer;
it is
Sylvester’s
Law of Inertia
treated elsewhere in this course.
The usual notation for complex vector spaces differs slightly from the notation for real spaces.
Linear functionals are written
w
H
or
w
*
instead of
w
T
,
and row vectors are written
w
H
or
w*
to denote the complex conjugate transpose of column
w
instead of merely its transpose
w
T
.
( Matlab
uses
“
w.’
”
for
w
T
and
“
w’
”
for
w* .)
We’ll use the
w*
notation because
it is older and more widespread than
w
H
.
Matrix
T
is congruent to
C*TC
whenever
C
is
any invertible matrix and
C*
is its complex conjugate transpose.
Most theorems are the same
for complex as for real spaces;
for instance
Sylvester’s Law of Inertia
holds for congruences
among complex
Hermitian
matrices
T = T*
as well as real symmetric.
Because many proofs
are simpler for real spaces we shall stay mostly with them.
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 Fall '08
 Chandrasekara
 Linear Algebra, Norms, Prof. W. Kahan

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