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Cauchy–Schwarz inequality. The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) [1][2][3][4] is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.
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 ...
In mathematics, the rearrangement inequality[1] states that for every choice of real numbers and every permutation of the numbers we have. . (1) Informally, this means that in these types of sums, the largest sum is achieved by pairing large values with large values, and the smallest sum is achieved by pairing small values with large values ...
Viktor Bunyakovsky. Baron Augustin-Louis Cauchy FRS FRSE (UK: / ˈkoʊʃi / KOH-shee, / ˈkaʊʃi / KOW-shee, [1][2] US: / koʊˈʃiː / koh-SHEE; [2][3] French: [oɡystɛ̃ lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of ...
In probability theory and statistics, a complex random vector is typically a tuple of complex -valued random variables, and generally is a random variable taking values in a vector space over the field of complex numbers. If are complex-valued random variables, then the n -tuple is a complex random vector. Complex random variables can always be ...
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 .
Polarization identity. In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in ...
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.