Home

PxQx

PxQx is a compact notation used in certain areas of linear algebra and functional analysis to denote the sequential action of two linear operators on a vector x. In this usage, P and Q are linear operators on a vector space V, and PxQx is defined as Q(P(x)); that is, first apply P to x, then apply Q to the result. The expression is read as “Q after P on x.” This notation is not universally standard, but it appears in lecture notes and expository texts as a concise way to discuss operator composition and projection concepts.

Formal definition and properties

Let P and Q be linear operators on V. PxQx is the result of applying P to

Examples

In R^n, suppose P is the orthogonal projection onto a subspace A and Q is the projection

Applications

PxQx serves as a mnemonic in discussions of operator composition, projection methods, and the study of subspace

See also

Composition of linear operators, projection (linear algebra), alternating projections.

x
and
then
applying
Q
to
the
outcome,
so
PxQx
=
Q(P(x)).
If
P
and
Q
are
projections
(idempotent
operators
with
P^2
=
P
and
Q^2
=
Q),
the
behavior
of
PxQx
depends
on
whether
P
and
Q
commute.
If
PQ
=
QP,
then
PxQx
=
PQx,
a
simplification
useful
in
proofs
and
examples.
If
they
do
not
commute,
PxQx
cannot
generally
be
rewritten
in
terms
of
a
single
projection
without
additional
structure.
onto
subspace
B.
Then
PxQx
equals
the
projection
onto
B
of
the
projection
onto
A
of
x.
If
A
⊆
B,
PxQx
simplifies
to
Qx;
if
the
subspaces
are
disjoint
or
in
general
position,
PxQx
reflects
the
interaction
of
the
two
projections.
interactions.
It
is
especially
relevant
in
theoretical
contexts
and
in
explanations
of
how
consecutive
projections
transform
a
vector.