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Projection

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In linear algebra, we deal with the vector, vector spaces, linear equations etc. We study about matrices quite frequently in linear algebra. There are various different concepts based on matrices and are studied in this branch of mathematics.

The projection is one of important concepts. The projection is broadly said to be the transformation of lines or points from one plane onto some other plane. It is done by connecting the given points located on two planes with the help of a set of parallel lines. The branch of geometry which studies about the projection of geometrical figure and the invariant and properties of projection, is known as projective geometry.
In this page, we are going to learn about the projection of a matrix. The matrices may represent linear equations. The projection of a matrix may denote the projection of a system of linear equations in one plane onto another plane. So, lets go ahead and understand about the concept of projection of matrices in this page below.

Definition

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In functional analysis and linear algebra, the projection is defined as a linear transformation from some vector space onto itself in such a way that if the transformation is applied two times to some value, then it will provide the same value.

Let u suppose that there is a linear transformation P which is defined from a vector space (say V) to itself, i.e.

V : P $\rightarrow$ P, such that

P$^{2}$ = P

We can say that when P is applied twice, we get the the same result as if P were applied only once (known as idempotent). This process provides its unchanged image.

For Example:

Let  $A = \begin{bmatrix}
0 & 0\\
k & 1
\end{bmatrix}$

Then,

$A^{2} = \begin{bmatrix}
0 & 0\\
k & 1
\end{bmatrix} \begin{bmatrix}
0 & 0\\
k & 1
\end{bmatrix} = \begin{bmatrix}
0 & 0\\
k & 1
\end{bmatrix}$

Therefore,

$A^{2} = A$

which means that A is a projection.

Properties

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A projection matrix has many properties, some important ones are discussed below. Let us assume that V be a vector space of finite dimensions. Also, P be the projection on V. Consider that R and K be the subspaces of V and are the range and kernel of P respectively.

Then, the basic properties followed by the projection P are listed below:

1) P must be idempotent, i.e. P$^{2}$ = P.

2) P is symmetric matrix.

3) P is defined as an identity operator I on over the range R, i.e.
for every x $\in$ R, Px = x

4) Each vector x in vector space V could be uniquely decomposed as the sum x = r + k such that u = Px and v = x − Px = (I − P)x, where r $\in$ R and k $\in$ K.

5) The range and kernel of a projection P are complementary to each other and are denoted by R and K = I - R or K and R = I - K.

Applications

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The projections are widely applicable in linear algebra. They play a significant role in algorithms of many problems in linear algebra. Some important ones are QR decomposition, reduction to Hessenberg form, singular value decomposition, linear regression etc. The projections are utilized as the special cases of idempotents. These idempotents are being used in the classification such as semisimple algebras. Therefore, we may imagine projections as very often encountered in the concepts of operator algebras.