Derivations for Linear Algebra and Optimization
نویسنده
چکیده
Much of this section was copied and paraphrased from Heath’s Scientific Computing. Anyways. Suppose we are looking for an orthogonal transformation that annihilates desired components of a given vector. Recall that a square real matrix Q is said to be orthogonal if its columns are orthonormal, that is, that Q Q = I. Orthogonal transformations are nice because they preserve Euclidean norms of any vectors v, as follows: ‖Qv‖2 = (Qv) Qv = v Q Qv = v v = ‖v‖2 These sorts of transformations are nice, because when applied to any linear system, they maintain Euclidean norms and won’t penalize the difficulty of solving the problem numerically. Now, we are really looking for an orthogonal transformation of a vector that annihilates its components while maintaining the two-norm (or Euclidean norm) ‖·‖2. We can accomplish this with a Householder transformation, which is a matrix of the form H = I − 2 T vT v .
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