Linear Algebra/Orthogonality

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Cauchy-Schwarz inequality[edit | edit source]

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[edit | edit source]

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

Orthogonal complement and matrix transpose[edit | edit source]

Applications[edit | edit source]

Linear least squares[edit | edit source]

How to orthogonalize a basis[edit | edit source]

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.