Thread: orthogonal projection

Let T be a linear operator on a finite-dimensional inner product space V.
1. If T is an orthogonal projection, prove that ||T(x)|| ||x|| for all x V. Give an example of a projection for which this inequality does not hold. What can be concluded about a projection for which the inequality is actually an equality for all x V?

2. Suppose that T is a projection such that ||T(x)|| ||x|| for x V. Prove that T is an orthogonal projection.

Originally Posted by studentmath92

Let T be a linear operator on a finite-dimensional inner product space V.
1. If T is an orthogonal projection, prove that ||T(x)|| ||x|| for all x V. Give an example of a projection for which this inequality does not hold. What can be concluded about a projection for which the inequality is actually an equality for all x V?

2. Suppose that T is a projection such that ||T(x)|| ||x|| for x V. Prove that T is an orthogonal projection.

For a (non-orthogonal) projection T and a vector x with , you could take and in the space .

For problem 2, if T is not orthogonal then the kernel of T is not orthogonal to the image of T. So there is a vector with . Then . But (because T is a projection). By Pythagoras' theorem, , and so .