The Cauchy Inequality, also known as the Cauchy–Schwarz Inequality, is one of the most important inequalities in algebra and mathematics.
Statement of the Inequality
For any real numbers \( a_1, a_2, \dots, a_n \) and \( b_1, b_2, \dots, b_n \),
\[ (a_1b_1 + a_2b_2 + \dots + a_nb_n)^2 \leq (a_1^2 + a_2^2 + \dots + a_n^2) (b_1^2 + b_2^2 + \dots + b_n^2) \]
In simple words:
The square of the sum of products is less than or equal to the product of the sums of squares.
The square of the sum of products is less than or equal to the product of the sums of squares.
Two-Variable Form
For two numbers, it becomes:
\[ (a_1b_1 + a_2b_2)^2 \leq (a_1^2 + a_2^2)(b_1^2 + b_2^2) \]Simple Example
Let
\[ a_1 = 1, \quad a_2 = 2 \] \[ b_1 = 3, \quad b_2 = 4 \]Left side:
\[ (1\cdot3 + 2\cdot4)^2 = (3 + 8)^2 = 11^2 = 121 \]Right side:
\[ (1^2 + 2^2)(3^2 + 4^2) = (1 + 4)(9 + 16) = (5)(25) = 125 \]Since \( 121 \leq 125 \), the inequality is true.
When Does Equality Occur?
Equality happens when the two sets of numbers are proportional, meaning:
\[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \dots = \frac{a_n}{b_n} \]In simple terms, one list is a constant multiple of the other.
Why Is It Important?
- Used in algebra and inequalities
- Important in vector geometry
- Fundamental in linear algebra
- Used in probability and statistics
Final Takeaway
\[ \left( \sum_{i=1}^{n} a_i b_i \right)^2 \leq \left( \sum_{i=1}^{n} a_i^2 \right) \left( \sum_{i=1}^{n} b_i^2 \right) \]Cauchy’s Inequality gives a powerful way to compare sums and products, and it appears everywhere in higher mathematics.