Difference between revisions of "Cauchy-schwarz inequality"

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(If you say so.)
 
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Consider the [[quadratic]],
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#REDIRECT [[Cauchy-Schwarz Inequality]]
<math>(a_1x+b_1)^2+(a_2x+b_2)^2+...(a_nx+b_n)^2 = 0</math>.
 
Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math>
 
Since the equation is always greater than or equal to 0, <math>B^2-4AC \leq 0</math>.  Substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', which states that
 
<math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>,
 
or, in the more compact [[sigma notation]],
 
<math>\left(\sum a_ib_i\right) \leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math>
 
''This page should be deleted as it has become obsolete with the more complete [[Cauchy-Schwarz Inequality]]'' page with a capital S in shwarz"
 

Latest revision as of 17:22, 18 June 2006