Linear Algebra/Inner Product Length and Orthogonality
Orthogonality edit
Cauchy-Schwarz inequality edit
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
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
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-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.