Editing Projection (linear algebra) (section)

===Spectrum===
In infinite-dimensional vector spaces, the [[Spectrum (functional analysis)|spectrum]] of a projection is contained in <math>\{ 0, 1 \}</math> as
<math display="block">(\lambda I - P)^{-1} = \frac 1 \lambda I + \frac 1 {\lambda(\lambda-1)} P.</math>
Only 0 or 1 can be an [[eigenvalue]] of a projection. This implies that an orthogonal projection <math>P</math> is always a [[positive semi-definite matrix]]. In general, the corresponding [[eigenspace]]s are (respectively) the kernel and range of the projection. Decomposition of a vector space into direct sums is not unique. Therefore, given a subspace <math>V</math>, there may be many projections whose range (or kernel) is <math>V</math>.

If a projection is nontrivial it has [[minimal polynomial (linear algebra)|minimal polynomial]] <math>x^2 - x = x (x-1)</math>, which factors into distinct linear factors, and thus <math>P</math> is [[diagonalizable]].