Linear Algebra/Orthogonality

< Linear algebra

Cauchy-Schwarz inequality

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The Cauchy-Schwarz inequality states that the magnitude of the inner product of two vectors is less than or equal to the product of the vector norms, or:  .

Definition

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For any vectors   and   in an inner product space  , we say   is orthogonal to  , and denote it by  , if  .

Orthogonal complement and matrix transpose

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Applications

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Linear least squares

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How to orthogonalize a basis

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Suppose to be on a vector space V with a scalar product (not necessarily positive-definite),
Problem: Construct an orthonormal basis of V starting by a random basis { v1, ... }.
Solution: Gram-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.