Schwarz Inequality, also known as Cauchy–Schwarz inequality, Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky inequality, is useful in many Mathematical fields such as Linear Algebra. This Inequality was formulated by Augustin Cauchy (1821), Viktor Yakovlevich Bunyakovsky (1859) and Hermann Amandus Schwarz (1888).
The uncertainty principle of quantum mechanics, which relates the incompatibility of two operators, rests on this important theorem of Schwarz.
This is a theorem that arise from the inner product of two vectors which sates that the square magnitude of the inner product of two vectors is less than or equal to the product of the square magnitude of any vector, i. e.,
where α and β are scalar constants.
By going through the derivation of Schwarz Inequality, show that the inequality becomes an equality if
where μ is an arbitrary constant.
Starting with the Schwarz Inequality
with the general equation
From the axiom;
we let the axiom equal to zero and substitute the value V so then we have,
Doing algebra and simple transformation we arrive to the equation
and from the general equation we have, we derived this
with the condition
About the Author
Debbie Claire R. Sanchez is currently a student of MSU-IIT pursuing her graduate study and hopefully will be graduating soon. She is very much interested in the field of Materials Science more specifically on Polymers. She plans to pursue her Ph. D in the United States and dreams on working in a well known company.