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

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

Thus Thus, we have proved the following theorem.

- the matrix is symmetric,
- and

- Show that for any non-zero vector the rank of the matrix is
- 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
- 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
- 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.
- 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
- prove that is a linear transformation,
- find the matrix, and
- prove that is an orthogonal matrix.