Cauchy–Schwarz inequality

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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.[5]

Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was published by Viktor Bunyakovsky (1859)[2] and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version.[5]

Statement of the inequality

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The Cauchy–Schwarz inequality states that for all vectors   and   of an inner product space

  (1)

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). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:[6][7]

  (2)

Moreover, the two sides are equal if and only if   and   are linearly dependent.[8][9][10]

Special cases

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Sedrakyan's lemma - Positive real numbers

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Sedrakyan's inequality, also known as Bergström's inequality, Engel's form, Titu's lemma (or the T2 lemma), states that for real numbers   and positive real numbers  :  

It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the dot product on   upon substituting   and  . This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.

R2 - The plane

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Cauchy-Schwarz inequality in a unit circle of the Euclidean plane

The real vector space   denotes the 2-dimensional plane. It is also the 2-dimensional Euclidean space where the inner product is the dot product. If   and   then the Cauchy–Schwarz inequality becomes:   where   is the angle between   and  .

The form above is perhaps the easiest in which to understand the inequality, since the square of the cosine can be at most 1, which occurs when the vectors are in the same or opposite directions. It can also be restated in terms of the vector coordinates  ,  ,  , and   as   where equality holds if and only if the vector   is in the same or opposite direction as the vector  , or if one of them is the zero vector.

Rn: n-dimensional Euclidean space

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In Euclidean space   with the standard inner product, which is the dot product, the Cauchy–Schwarz inequality becomes:  

The Cauchy–Schwarz inequality can be proved using only elementary algebra in this case by observing that the difference of the right and the left hand side is  

or by considering the following quadratic polynomial in    

Since the latter polynomial is nonnegative, it has at most one real root, hence its discriminant is less than or equal to zero. That is,  

Cn: n-dimensional Complex space

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If   with   and   (where   and  ) and if the inner product on the vector space   is the canonical complex inner product (defined by   where the bar notation is used for complex conjugation), then the inequality may be restated more explicitly as follows:  

That is,  

For the inner product space of square-integrable complex-valued functions, the following inequality holds.  

The Hölder inequality is a generalization of this.

Applications

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Analysis

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In any inner product space, the triangle inequality is a consequence of the Cauchy–Schwarz inequality, as is now shown:  

Taking square roots gives the triangle inequality:  

The Cauchy–Schwarz inequality is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.[11][12]

Geometry

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The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining:[13][14]  

The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval [−1, 1] and justifies the notion that (real) Hilbert spaces are simply generalizations of the Euclidean space. It can also be used to define an angle in complex inner-product spaces, by taking the absolute value or the real part of the right-hand side,[15][16] as is done when extracting a metric from quantum fidelity.

Probability theory

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Let   and   be random variables. Then the covariance inequality[17][18] is given by:  

After defining an inner product on the set of random variables using the expectation of their product,   the Cauchy–Schwarz inequality becomes  

To prove the covariance inequality using the Cauchy–Schwarz inequality, let   and   then   where   denotes variance and   denotes covariance.

Proofs

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There are many different proofs[19] of the Cauchy–Schwarz inequality other than those given below.[5][7] When consulting other sources, there are often two sources of confusion. First, some authors define ⟨⋅,⋅⟩ to be linear in the second argument rather than the first. Second, some proofs are only valid when the field is   and not  [20]

This section gives two proofs of the following theorem:

Cauchy–Schwarz inequality — Let   and   be arbitrary vectors in an inner product space over the scalar field   where   is the field of real numbers   or complex numbers   Then

  (Cauchy–Schwarz Inequality)

with equality holding in the Cauchy–Schwarz Inequality if and only if   and   are linearly dependent.

Moreover, if   and   then  


In both of the proofs given below, the proof in the trivial case where at least one of the vectors is zero (or equivalently, in the case where  ) is the same. It is presented immediately below only once to reduce repetition. It also includes the easy part of the proof of the Equality Characterization given above; that is, it proves that if   and   are linearly dependent then  

Proof of the trivial parts: Case where a vector is   and also one direction of the Equality Characterization

By definition,   and   are linearly dependent if and only if one is a scalar multiple of the other. If   where   is some scalar then  

which shows that equality holds in the Cauchy–Schwarz Inequality. The case where   for some scalar   follows from the previous case:  

In particular, if at least one of   and   is the zero vector then   and   are necessarily linearly dependent (for example, if   then   where  ), so the above computation shows that the Cauchy-Schwarz inequality holds in this case.

Consequently, the Cauchy–Schwarz inequality only needs to be proven only for non-zero vectors and also only the non-trivial direction of the Equality Characterization must be shown.

Proof via the Pythagorean theorem

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The special case of   was proven above so it is henceforth assumed that   Let  

It follows from the linearity of the inner product in its first argument that:  

Therefore,   is a vector orthogonal to the vector   (Indeed,   is the projection of   onto the plane orthogonal to  ) We can thus apply the Pythagorean theorem to   which gives  

The Cauchy–Schwarz inequality follows by multiplying by   and then taking the square root. Moreover, if the relation   in the above expression is actually an equality, then   and hence   the definition of   then establishes a relation of linear dependence between   and   The converse was proved at the beginning of this section, so the proof is complete.  

Proof by analyzing a quadratic

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Consider an arbitrary pair of vectors  . Define the function   defined by  , where   is a complex number satisfying   and  . Such an   exists since if   then   can be taken to be 1.

Since the inner product is positive-definite,   only takes non-negative real values. On the other hand,   can be expanded using the bilinearity of the inner product:   Thus,   is a polynomial of degree   (unless   which is a case that was checked earlier). Since the sign of   does not change, the discriminant of this polynomial must be non-positive:   The conclusion follows.[21]

For the equality case, notice that   happens if and only if   If   then   and hence  

Generalizations

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Various generalizations of the Cauchy–Schwarz inequality exist. Hölder's inequality generalizes it to   norms. More generally, it can be interpreted as a special case of the definition of the norm of a linear operator on a Banach space (Namely, when the space is a Hilbert space). Further generalizations are in the context of operator theory, e.g. for operator-convex functions and operator algebras, where the domain and/or range are replaced by a C*-algebra or W*-algebra.

An inner product can be used to define a positive linear functional. For example, given a Hilbert space   being a finite measure, the standard inner product gives rise to a positive functional   by   Conversely, every positive linear functional   on   can be used to define an inner product   where   is the pointwise complex conjugate of   In this language, the Cauchy–Schwarz inequality becomes[22]  

which extends verbatim to positive functionals on C*-algebras:

Cauchy–Schwarz inequality for positive functionals on C*-algebras[23][24] — If   is a positive linear functional on a C*-algebra   then for all    

The next two theorems are further examples in operator algebra.

Kadison–Schwarz inequality[25][26] (Named after Richard Kadison) — If   is a unital positive map, then for every normal element   in its domain, we have   and  

This extends the fact   when   is a linear functional. The case when   is self-adjoint, that is,   is sometimes known as Kadison's inequality.

Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps[27]) — For a 2-positive map   between C*-algebras, for all   in its domain,    

Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:

Callebaut's Inequality[28] — For reals    

This theorem can be deduced from Hölder's inequality.[29] There are also non-commutative versions for operators and tensor products of matrices.[30]

Several matrix versions of the Cauchy–Schwarz inequality and Kantorovich inequality are applied to linear regression models.[31] [32]

See also

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Notes

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Citations

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  1. ^ O'Connor, J.J.; Robertson, E.F. "Hermann Amandus Schwarz". University of St Andrews, Scotland.
  2. ^ a b Bityutskov, V. I. (2001) [1994], "Bunyakovskii inequality", Encyclopedia of Mathematics, EMS Press
  3. ^ Ćurgus, Branko. "Cauchy-Bunyakovsky-Schwarz inequality". Department of Mathematics. Western Washington University.
  4. ^ Joyce, David E. "Cauchy's inequality" (PDF). Department of Mathematics and Computer Science. Clark University. Archived (PDF) from the original on 2022-10-09.
  5. ^ a b c Steele, J. Michael (2004). The Cauchy–Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities. The Mathematical Association of America. p. 1. ISBN 978-0521546775. ...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.
  6. ^ Strang, Gilbert (19 July 2005). "3.2". Linear Algebra and its Applications (4th ed.). Stamford, CT: Cengage Learning. pp. 154–155. ISBN 978-0030105678.
  7. ^ a b Hunter, John K.; Nachtergaele, Bruno (2001). Applied Analysis. World Scientific. ISBN 981-02-4191-7.
  8. ^ Bachmann, George; Narici, Lawrence; Beckenstein, Edward (2012-12-06). Fourier and Wavelet Analysis. Springer Science & Business Media. p. 14. ISBN 9781461205050.
  9. ^ Hassani, Sadri (1999). Mathematical Physics: A Modern Introduction to Its Foundations. Springer. p. 29. ISBN 0-387-98579-4. Equality holds iff <c|c>=0 or |c>=0. From the definition of |c>, we conclude that |a> and |b> must be proportional.
  10. ^ Axler, Sheldon (2015). Linear Algebra Done Right, 3rd Ed. Springer International Publishing. p. 172. ISBN 978-3-319-11079-0. This inequality is an equality if and only if one of u, v is a scalar multiple of the other.
  11. ^ Bachman, George; Narici, Lawrence (2012-09-26). Functional Analysis. Courier Corporation. p. 141. ISBN 9780486136554.
  12. ^ Swartz, Charles (1994-02-21). Measure, Integration and Function Spaces. World Scientific. p. 236. ISBN 9789814502511.
  13. ^ Ricardo, Henry (2009-10-21). A Modern Introduction to Linear Algebra. CRC Press. p. 18. ISBN 9781439894613.
  14. ^ Banerjee, Sudipto; Roy, Anindya (2014-06-06). Linear Algebra and Matrix Analysis for Statistics. CRC Press. p. 181. ISBN 9781482248241.
  15. ^ Valenza, Robert J. (2012-12-06). Linear Algebra: An Introduction to Abstract Mathematics. Springer Science & Business Media. p. 146. ISBN 9781461209010.
  16. ^ Constantin, Adrian (2016-05-21). Fourier Analysis with Applications. Cambridge University Press. p. 74. ISBN 9781107044104.
  17. ^ Mukhopadhyay, Nitis (2000-03-22). Probability and Statistical Inference. CRC Press. p. 150. ISBN 9780824703790.
  18. ^ Keener, Robert W. (2010-09-08). Theoretical Statistics: Topics for a Core Course. Springer Science & Business Media. p. 71. ISBN 9780387938394.
  19. ^ Wu, Hui-Hua; Wu, Shanhe (April 2009). "Various proofs of the Cauchy-Schwarz inequality" (PDF). Octogon Mathematical Magazine. 17 (1): 221–229. ISBN 978-973-88255-5-0. ISSN 1222-5657. Archived (PDF) from the original on 2022-10-09. Retrieved 18 May 2016.
  20. ^ Aliprantis, Charalambos D.; Border, Kim C. (2007-05-02). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer Science & Business Media. ISBN 9783540326960.
  21. ^ Rudin, Walter (1987) [1966]. Real and Complex Analysis (3rd ed.). New York: McGraw-Hill. ISBN 0070542341.
  22. ^ Faria, Edson de; Melo, Welington de (2010-08-12). Mathematical Aspects of Quantum Field Theory. Cambridge University Press. p. 273. ISBN 9781139489805.
  23. ^ Lin, Huaxin (2001-01-01). An Introduction to the Classification of Amenable C*-algebras. World Scientific. p. 27. ISBN 9789812799883.
  24. ^ Arveson, W. (2012-12-06). An Invitation to C*-Algebras. Springer Science & Business Media. p. 28. ISBN 9781461263715.
  25. ^ Størmer, Erling (2012-12-13). Positive Linear Maps of Operator Algebras. Springer Monographs in Mathematics. Springer Science & Business Media. ISBN 9783642343698.
  26. ^ Kadison, Richard V. (1952-01-01). "A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras". Annals of Mathematics. 56 (3): 494–503. doi:10.2307/1969657. JSTOR 1969657.
  27. ^ Paulsen, Vern (2002). Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics. Vol. 78. Cambridge University Press. p. 40. ISBN 9780521816694.
  28. ^ Callebaut, D.K. (1965). "Generalization of the Cauchy–Schwarz inequality". J. Math. Anal. Appl. 12 (3): 491–494. doi:10.1016/0022-247X(65)90016-8.
  29. ^ Callebaut's inequality. Entry in the AoPS Wiki.
  30. ^ Moslehian, M.S.; Matharu, J.S.; Aujla, J.S. (2011). "Non-commutative Callebaut inequality". Linear Algebra and Its Applications. 436 (9): 3347–3353. arXiv:1112.3003. doi:10.1016/j.laa.2011.11.024. S2CID 119592971.
  31. ^ Liu, Shuangzhe; Neudecker, Heinz (1999). "A survey of Cauchy-Schwarz and Kantorovich-type matrix inequalities". Statistical Papers. 40: 55–73. doi:10.1007/BF02927110. S2CID 122719088.
  32. ^ Liu, Shuangzhe; Trenkler, Götz; Kollo, Tõnu; von Rosen, Dietrich; Baksalary, Oskar Maria (2023). "Professor Heinz Neudecker and matrix differential calculus". Statistical Papers. 65 (4): 2605–2639. doi:10.1007/s00362-023-01499-w. S2CID 263661094.

References

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