16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.3.11

Operators on Inner Product Spaces

Mathematics — Linear Algebra · positive-operators-projection

Problem

Let

V

be a finite-dimensional inner product space and

E

the orthogonal projection of

V

onto some subspace. (a) Prove that, for any positive number

c

, the operator

cI + E

is positive. (b) Express in terms of

E

a self-adjoint linear operator

T

such that

T^2 = I + E