| Orthogonal Projection |
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Information AboutOrthogonal Projection |
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If such a projection leaves the origin Fixed , it is a Self-adjoint Idempotent Linear Transformation ; its Matrix is a Symmetric idempotent matrix. Conversely, every symmetric idempotent matrix is the matrix of the orthogonal projection onto its own column space. If ''M'' is a ''k''×''d'' matrix with ''d'' < ''k'', the ''d'' columns spanning a ''d''-dimensional subspace, then the matrix of the orthogonal projection onto the column space of ''M'' is : (and before leaping to the conclusion that this must be an identity matrix, remember that ''M'' is not a square matrix but has more rows than columns!). If the Basis is Orthonormal , the projection can be simplified to : In Functional Analysis , the geometric notion is generalized as follows. An orthogonal projection is a Bounded Operator on a Hilbert Space H which is Self-adjoint and Idempotent . It maps each Vector ''v'' in ''H'' to the closest point of ''PH'' to ''v''. ''PH'' is the Range of ''P'' and it is a Closed Subspace of ''H''. See also Spectral Theorem , Orthogonal Matrix . TECHNICAL DRAWING A concrete instance is used in Technical Drawing , where orthogonal projection, more correctly called Orthographic Projection . Views of an object projected onto orthogonal planes. Commonly known views of this type are ''plan'' ('' Plan View ''), '' Side View '' and '' Elevation ''. |
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