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.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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.)
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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
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