Let V be finite-dimensional and let P = L(V) be such that P² = P. Suppose P is self-adjoint. Show that there exists a subspace U of V such that P = Pru. (Note: say that orthogonal projections are the same as projections that are also self-adjoint.)
Let V be finite-dimensional and let P = L(V) be such that P² = P. Suppose P is self-adjoint. Show that there exists a subspace U of V such that P = Pru. (Note: say that orthogonal projections are the same as projections that are also self-adjoint.)
Let V be finite-dimensional and let P = L(V) be such that P² = P. Suppose P is self-adjoint. Show that there exists a subspace U of V such that P = Pru. (Note: say that orthogonal projections are the same as projections that are also self-adjoint.)
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
Expert Solution
Introduction
As per the question we are given a finite-dimensional vector space V and P ∈ L(V) be such that P2 = P, such that P is self-adjoint.
Now we have to show that there exists a subspace U of V such that P = PrU