Cauchy-Schwarz Inequality
Learn the Cauchy-Schwarz inequality with proofs and examples for vectors, sums, and integrals in mathematics.
The Formula
For finite sums: (∑ aibi)² ≤ (∑ ai²)(∑ bi²)
The Cauchy-Schwarz inequality is one of the most important and widely used inequalities in all of mathematics. It states that for any two vectors in an inner product space, the absolute value of their inner product is at most equal to the product of their norms. Equality holds if and only if the two vectors are linearly dependent (one is a scalar multiple of the other).
In its simplest form for real numbers, if you have two lists of numbers (a1, a2, ..., an) and (b1, b2, ..., bn), then the square of their dot product is less than or equal to the product of the sum of squares of each list. This can be written as (a1b1 + a2b2 + ... + anbn)² ≤ (a1² + a2² + ... + an²)(b1² + b2² + ... + bn²).
The inequality was first published by Augustin-Louis Cauchy in 1821 in France for finite sums, later extended to integrals by Viktor Bunyakovsky in 1859 in Russia, and further generalized by Hermann Amandus Schwarz in 1885 in Germany. In some references, it is called the Cauchy-Bunyakovsky-Schwarz inequality.
The geometric interpretation is intuitive. The dot product of two vectors equals the product of their magnitudes times the cosine of the angle between them. Since cosine ranges from −1 to 1, the absolute value of the dot product cannot exceed the product of the magnitudes. This is exactly what the Cauchy-Schwarz inequality states.
This inequality serves as the foundation for many other results, including the triangle inequality for norms, the concept of angles between vectors in abstract spaces, and the Pearson correlation coefficient being bounded between −1 and 1.
Variables
| Symbol | Meaning |
|---|---|
| ⟨u, v⟩ | Inner product (dot product) of vectors u and v |
| ai, bi | Components of two real-valued sequences |
| n | Number of components in each sequence |
| ||u|| | Norm (magnitude) of vector u, equal to √⟨u, u⟩ |
Example 1
Verify the Cauchy-Schwarz inequality for a = (1, 2, 3) and b = (4, 5, 6).
Dot product: a · b = 1(4) + 2(5) + 3(6) = 4 + 10 + 18 = 32
Left side: (a · b)² = 32² = 1024
|a|² = 1 + 4 + 9 = 14 and |b|² = 16 + 25 + 36 = 77
Right side: 14 × 77 = 1078
1024 ≤ 1078. The inequality holds. Since 1024 ≠ 1078, the vectors are not parallel.
Example 2
Show that equality holds for a = (2, 4) and b = (3, 6).
Note that b = 1.5 × a (they are proportional)
Dot product: 2(3) + 4(6) = 6 + 24 = 30
Left side: 30² = 900
|a|² = 4 + 16 = 20 and |b|² = 9 + 36 = 45
Right side: 20 × 45 = 900
900 = 900. Equality holds because b is a scalar multiple of a.
When to Use It
The Cauchy-Schwarz inequality is used across many fields of mathematics and science.
- Proving the triangle inequality and other fundamental inequalities
- Statistics: showing that the correlation coefficient lies between −1 and 1
- Quantum mechanics: proving the uncertainty principle
- Machine learning: bounding cosine similarity between feature vectors
- Optimization problems and mathematical proofs involving sums or integrals
- Signal processing: comparing signals using inner products