STAT312 Lecture Notes - Global Positioning System, Projection Matrix, Idempotent Matrix
5 Apr 2014
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Document Summary
The trace of a square matrix is the sum of its diagonal elements. A useful identity is (how?) (ab) = (ba) Thus products within traces can be rearranged cyclically: (abc) = (cab) = but not necessarily = (bca) (acb) It will be shown that for an idempotent matrix, A consequence is that dim (x) = (i h) (i h) (h) A matrix q is orthogonal if the columns are mutually orthogonal, and have unit norm. If q is orthogonal then kqyk = kyk for any. 1 vector y norms are preserved . Simi- larly, angles between vectors are also preserved (why?). Geometrically, an orthogonal transfor- mation is a rigid motion it corresponds to a rotation and/or an interchange of two or more axes. Gram-schmidt theorem: every has an orthogonal basis. vector space v . Proof: start with any basis v1 ize v1 to get a unit vector (i. e. a vector with unit norm) q1.
