Orthogonal Projection

linear-algebra

Definition

Orthogonal Projection

Let be a linear operator on an inner product space . The operator is an orthogonal projection if it is Hermitian and idempotent, that is,

Equivalently, projects each vector onto a subspace along its orthogonal complement.

A projection keeps one subspace and removes the orthogonal complement.

The Kept Subspace

The subspace that is kept is the image of the projection:

This means is the set of all possible outputs of . If , then for some . Since is idempotent,

So every vector in is fixed by . This is the sense in which keeps .

For an orthogonal projection, the removed directions are exactly perpendicular to . Those directions form the orthogonal complement .

Image and Kernel

Let . An orthogonal projection splits the space into two perpendicular parts:

Thus every decomposes uniquely as

with and . The projection acts by

So the sentence means:

The part in is fixed. The part in is sent to the zero vector.

Equivalently, every decomposes as

where and . Since is Hermitian, the kernel is the orthogonal complement of the image:

Thus keeps the component in its image (eigenvalue ) and removes the component in the orthogonal complement (eigenvalue ).

Eigenvalues

Eigenvalues of a Projection

Projections have eigenvalues and .

Eigenvalues of a Projection

Let be a non-zero vector and be a projection

Suppose

Apply again:

Since ,

Therefore:

Since ,

So:

Therefore:

Example

Projection onto one basis state

Let be a normalised vector in an inner product space, so

The operator

is an orthogonal projection onto the one-dimensional subspace .

Acting on a vector , it gives

Thus keeps the component of in the direction and removes the orthogonal component.

It is idempotent:

It is also Hermitian:

Therefore is an orthogonal projection.

Example

The operator , given by the matrix:

is an orthogonal projection.

In particular, corresponds to the projection to the first component: