# Direct Sum Decompositions

Recall that subspaces*X*and

*Y*of a vector space over field 𝔽 (which is either ℚ or ℝ or ℂ) form a direct sum, denoted

*V*=

*X*⊕

*Y*, if for every

**v**∈

*V*there exist unique vectors

**x**∈

*X*and

**y**∈

*Y*such that

**v**=

**x**+

**y**.

Suppose that

From this definition, it follows
*V*=*X*⊕*Y*is a direct sum of two subspaces. Define*P*:*V*⇾*V*as follows. For**v**∈*V*write**v**=**x**+**y**, where**x**∈*X*and**y**∈*Y*, then define*P*(**v**) :=**x**.-
*P*is well-defined; -
*P*is a linear transformation; -
Im
*P*=*X*, Ker*P*=*Y*; -
*P*² =*P*.

A linear operator

*P*:*V*⇾*V*such that*P*² =*P*is called the**projection**or idempotent operator.*P*is also said to be the projection onto*X*along*Y*.
Example 1:
Consider the vertical projection

Suppose that *P*on the plane of equation*z*=*ax*+*by*in the usual ℝ³. This is a linear map and we can determine its matrix in the canonical basis {**i**,**j**,**k**} of ℝ³. End of Example 1

*V*is a finite-dimensional vecor space that is a direct sum of two subspaces,

*V*=

*X*⊕

*Y*. Choose α = {

**u**

_{1},

**u**

_{2}, … ,

**u**

_{r}}, a basis in

*X*and β = {

**w**

_{1},

**w**

_{2}, … ,

**w**

_{s}}, a basis in

*Y*. Then the matrix of

*P*with respect to the basis { α, β } is

\[
[\,P\,] = {\bf P} = \begin{bmatrix} {\bf I}_r & {\bf 0} \\ {\bf 0} & {\bf 0} \end{bmatrix} ,
\]

where **I**

_{r}is

*r*×

*r*identity matrix.

When dimension of Euclidean space *E* exceeds 1, the quadratic equation *P*² = *P* has infinitely many solutions *P* that are linear maps *E* ⇾ *E*. Recall that the quadratic equation
*X*² = −*I* also has infinitely many solutions *X*, which are 2 x 2 matrices.

Example 2:
We consider the space ℭ( ℝ) of continuous functions

*f*: ℝ ⇾ ℝ. For any such function, we define its symmetric part*f*_{s}by*f*_{s}(*x*) =*f*(−*x*), and consider the linear map:
\[
f \mapsto P\,f = \frac{1}{2} \left[ f(x) + f(-x) \right] .
\]

The image *g*=*Pf*is the average of*f*and its symmetric: It is an even function. Recall that even functions*g*are characterized by
\[
g(-x) = g(x) , \qquad x\in \mathbb{R} .
\]

For an even function *g*,*Pg*=*g*, and hence*P*²*f*=*Pg*=*g*=*Pf*. This proves*P*² =*P*so that*P*is a projector. In fact,*f*=*Pf*precisely when*f*is even, so that the image of*f*is the subspace*V*of even functions, on which*P*acts by the identity. On the other hand, if*h*is an odd function, namely,
\[
h(-x) = -h(x) , \qquad x\in \mathbb{R} ,
\]

then *Ph*= 0: The kernel of*P*is the subspace of odd functions. Any function*f*is the sum of an even one*g*=*Pf*and an odd one \( \displaystyle h = Q\, f = \frac{1}{2} \left[ f(x) - f(-x) \right] \) and
\[
P\,f = P \left( g+h \right) = P\,g + P\, h = P\,g = g.
\]

End of Example 2

**Theorem 1:**Let

*P*:

*V*⇾

*V*be the projection onto

*X*along

*Y*. Let

*T*:

*V*⇾

*V*be a linear transformation. Then

*PT*=

*TP*if and only if

*X*and

*Y*are

*T*-invariant, so

*T X*⊆

*X*and

*T Y*⊆

*Y*.

Example 3:

Let

*V*be a vector space over field 𝔽. Suppose that we have a finite set of projectors*P*_{i}:*V*⇾*V*,*i*= 1, 2, … ,*m*, such that*P*_{i}*P*_{j}= 0 whenever*i*≠*j*. If*P*_{1}+*P*_{2}+ ⋯ +*P*_{m}=*I*, the identity mapping, then the set of projectors {*P*_{1},*P*_{2}, … ,*P*_{m}} is known as a**partition of the identity**on*V*.**Theorem 2:**Suppose that a vector space

*V*is a direct sum of finite number of subspaces,

*V*=

*X*

_{1}⊕

*X*

_{2}⊕ ⋯ ⊕

*X*

_{m}. Let P

_{i}be the projection of

*V*onto

*X*

_{i}along its complement. Then the set of projectors {

*P*

_{1},

*P*

_{2}, … ,

*P*

_{m}} is a partition of the identity on

*V*. Conversely, if {

*P*

_{1},

*P*

_{2}, … ,

*P*

_{m}} is a partition of the identity on

*V*and

*X*

_{i}= Im

*P*

_{i}, then

*V*=

*X*

_{1}⊕

*X*

_{2}⊕ ⋯ ⊕

*X*

_{m}.

# Projection Operators

- Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.
- Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces (2nd ed.). Springer. ISBN 0-387-90093-4.