Difference between revisions of "Cauchy-schwarz inequality"
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<math>(a_1x+b_1)^2+(a_2x+b_2)^2+...(a_nx+b_n)^2 = 0</math>. | <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> | 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 \ | + | <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]], | or, in the more compact [[sigma notation]], | ||
| − | <math>\left(\sum a_ib_i\right) \ | + | <math>\left(\sum a_ib_i\right) \leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math> |
Revision as of 23:55, 17 June 2006
Consider the quadratic,
.
Expanding, we find the equation to be of the form 
, where 
, 
, and 
Since the equation is always greater than or equal to 0, 
. Substituting the above values of A, B, and C leaves us with the Cauchy-Schwarz Inequality, which states that
,
or, in the more compact sigma notation,
