Let P be the orthogonal projection onto a subspace E of an i

Let P be the orthogonal projection onto a subspace E of an inner product space V, dim V = n, dim E = r. Find the eigenvalues and the eigenvectors (eigenspaces). Find the algebraic and geometric multiplicities of each eigenvalue.

Solution

We can see that Pv=v if vE.

Now Pv=0 if vE, i.e, if v is orthogonal to every vector in E.

Then we can see that V=EE (in the informal sense), and we can use this to show that the only eigenvalues are 0 and 1, and the respective eigenspaces are E and E.

In other words orthogonal projection P from V onto the subspace E of V decomposes into an identity on E and 0 on E so the eigenvalues of P are 1 repeated dimension of E times and the rest are 0 repeated. the corresponding eigenvectors are the basis of E and E respectively.