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Cauchy's Inequality — Definition, Formula & Examples

Cauchy's Inequality (also called the Cauchy-Schwarz Inequality) states that the absolute value of the inner product of two vectors is at most the product of their magnitudes. In the finite-dimensional case, it says the square of a sum of products is less than or equal to the product of two sums of squares.

For real or complex sequences (a1,a2,…,an)(a_1, a_2, \ldots, a_n) and (b1,b2,…,bn)(b_1, b_2, \ldots, b_n), the inequality ∣∑k=1nakbk∣2≤(∑k=1n∣ak∣2)(∑k=1n∣bk∣2)\left|\sum_{k=1}^{n} a_k b_k\right|^2 \leq \left(\sum_{k=1}^{n} |a_k|^2\right)\left(\sum_{k=1}^{n} |b_k|^2\right) holds, with equality if and only if one sequence is a scalar multiple of the other.

Key Formula

(∑k=1nakbk) ⁣2≤(∑k=1nak2) ⁣(∑k=1nbk2)\left(\sum_{k=1}^{n} a_k b_k\right)^{\!2} \leq \left(\sum_{k=1}^{n} a_k^2\right)\!\left(\sum_{k=1}^{n} b_k^2\right)
Where:
  • aka_k = The k-th element of the first real sequence
  • bkb_k = The k-th element of the second real sequence
  • nn = The number of terms in each sequence

How It Works

To apply Cauchy's Inequality, identify two sequences or vectors and compute three sums: the sum of products ∑akbk\sum a_k b_k, the sum of squares ∑ak2\sum a_k^2, and the sum of squares ∑bk2\sum b_k^2. The inequality guarantees that the squared dot product never exceeds the product of the two sums of squares. Equality occurs precisely when ak=λbka_k = \lambda b_k for some constant λ\lambda and all kk. This makes the inequality a powerful tool for bounding expressions and proving that certain maxima or minima exist.

Worked Example

Problem: Verify Cauchy's Inequality for the vectors a = (1, 2, 3) and b = (4, 5, 6).
Compute the dot product: Multiply corresponding entries and sum.
∑akbk=1(4)+2(5)+3(6)=4+10+18=32\sum a_k b_k = 1(4) + 2(5) + 3(6) = 4 + 10 + 18 = 32
Compute the sums of squares: Find each sum of squares separately.
∑ak2=1+4+9=14,∑bk2=16+25+36=77\sum a_k^2 = 1 + 4 + 9 = 14, \qquad \sum b_k^2 = 16 + 25 + 36 = 77
Check the inequality: Square the dot product and compare it with the product of the sums of squares.
322=1024≤14×77=1078  ✓32^2 = 1024 \leq 14 \times 77 = 1078 \;\checkmark
Answer: Since 1024 ≤ 1078, Cauchy's Inequality holds. Equality does not occur because (1, 2, 3) is not a scalar multiple of (4, 5, 6).

Why It Matters

Cauchy's Inequality is a cornerstone in linear algebra, analysis, and probability. It underpins the definition of angle between vectors, the triangle inequality in normed spaces, and the proof that correlation coefficients lie between −1 and 1. You will encounter it repeatedly in courses on real analysis, functional analysis, and quantum mechanics.

Common Mistakes

Mistake: Forgetting to square the left-hand side and comparing the dot product directly to the product of magnitudes.
Correction: The inequality compares (∑akbk)2(\sum a_k b_k)^2 to (∑ak2)(∑bk2)\left(\sum a_k^2\right)\left(\sum b_k^2\right). Equivalently, you can compare ∣∑akbk∣|\sum a_k b_k| to ∑ak2 ∑bk2\sqrt{\sum a_k^2}\,\sqrt{\sum b_k^2}, but both sides must be treated consistently.