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where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
When , is a real number then the Cauchy–Schwarz inequality implies that , ‖ ‖ ‖ ‖ [,], and thus that (,) = , ‖ ‖ ‖ ‖, is a real number. This allows defining the (non oriented) angle of two vectors in modern definitions of Euclidean geometry in terms of linear algebra .
By the Cauchy–Schwarz inequality, ... as an array of complex numbers. ... the input space too difficult to work with in practice as properties of the original ...
Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, [4] this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce ...
1 Cauchy–Schwarz inequality 2 On a complex Hilbert space, if an operator is non-negative then it is symmetric 3 If an operator is non-negative and defined on the whole Hilbert space, then it is self-adjoint and bounded
The Cauchy–Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix G {\displaystyle G} are equal, which happens precisely when the vectors { x 1 , … , x m } {\displaystyle \{x_{1},\ldots ,x_{m ...
The feasible regions of linear programming are defined by a set of inequalities. In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size.
Cauchy's inequality may refer to: the Cauchy–Schwarz inequality in a real or complex inner product space Cauchy's estimate , also called Cauchy's inequality, for the Taylor series coefficients of a complex analytic function