## Matrix of the Orthogonal Projection

The minimization problem stated above arises in lot of applications. So, it will be very helpful if the matrix of the orthogonal projection can be obtained under a given basis.

To this end, let be a -dimensional subspace of with as its orthogonal complement. Let be the orthogonal projection of onto . Suppose, we are given an orthonormal basis of Under the assumption that is known, we explicitly give the matrix of with respect to an extended ordered basis of

Let us extend the given ordered orthonormal basis of to get an orthonormal ordered basis
of Then by Theorem 5.1.12, for any Thus, by definition, Let Consider the standard orthogonal ordered basis of Therefore, if for then

and

Then as observed in Remark 5.2.3.4, That is, for

 (5.3.1)

Thus, using the associativity of matrix product and (5.3.1), we get

Thus Thus, we have proved the following theorem.

THEOREM 5.3.15   Let be a -dimensional subspace of and let be the orthogonal projection of onto along Suppose, is an orthonormal ordered basis of Define an matrix Then the matrix of the linear transformation in the standard orthogonal ordered basis is

EXAMPLE 5.3.16   Let be a subspace of Then an orthonormal ordered basis of is

and that of is

Therefore, if is an orthogonal projection of onto along then the corresponding matrix is given by

Hence, the matrix of the orthogonal projection in the ordered basis

is

It is easy to see that
1. the matrix is symmetric,
2. and
Also, for any we have

Thus, is the closest vector to the subspace for any vector

EXERCISE 5.3.17
1. Show that for any non-zero vector the rank of the matrix is
2. Let be a subspace of a vector space and let be the orthogonal projection of onto along Let be an orthonormal ordered basis of Then prove that corresponding matrix satisfies
3. Let be an matrix with and Consider the associated linear transformation defined by for all Then prove that there exists a subspace of such that is the orthogonal projection of onto along
4. Let and be two distinct subspaces of a finite dimensional vector space Let and be the corresponding orthogonal projection operators of along and respectively. Then by constructing an example in show that the map is a projection but not an orthogonal projection.
5. Let be an -dimensional vector subspace of and let be its orthogonal complement. Let be an orthogonal ordered basis of with an ordered basis of Define a map

whenever for some and Then
1. prove that is a linear transformation,
2. find the matrix, and
3. prove that is an orthogonal matrix.
is called the reflection along

A K Lal 2007-09-12