Linear Algebra/Orthogonality

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Cauchy-Schwartz inequality[edit]

The Cauchy-Schwartz 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: | \langle x,y \rangle |  \le  \| x \| \| y \| .

Definition[edit]

For any vectors x and y in an inner product space V, we say x is orthogonal to y, and denote it by x \bot y, if \langle x,y \rangle =0.

Orthogonal complement and matrix transpose[edit]

Applications[edit]

Linear least squares[edit]

How to orthogonalize a basis[edit]

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-Schidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.