Cauchy-Schwarz Inequality¶

Theorem¶

For vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ in an inner product space, the Cauchy-Schwarz inequality states $|\langle\boldsymbol{x},\boldsymbol{y}\rangle|\leq\|\boldsymbol{x}\|\|\boldsymbol{y}\|$ $∥ x ∥∥ y ∥$ ; in $\mathbb{R}^n$ with the dot product this reads $|\boldsymbol{x}\cdot \boldsymbol{y}|\leq\|\boldsymbol{x}\|\|\boldsymbol{y}\|$ $∥ x ∥∥ y ∥$ .

Interpretation¶

The inner product of two vectors can never exceed the product of their lengths in absolute value. Equivalently, for nonzero vectors the quotient $\dfrac{\boldsymbol{x}\cdot \boldsymbol{y}}{\|\boldsymbol{x}\|\,\|\boldsymbol{y}\|}$ always lies between $-1$ and $1$ .

Two vectors x and y from the origin, with the projection of y onto x and an arc of radius equal to the length of y, showing that the projection is never longer than y.

The figure drops a perpendicular from the tip of $\boldsymbol{y}$ onto the line of the nonzero vector $\boldsymbol{x}$ . The green segment from the origin to its foot has length $\dfrac{|\boldsymbol{x}\cdot \boldsymbol{y}|}{\|\boldsymbol{x}\|}$ , and as a side of a right triangle with hypotenuse $\boldsymbol{y}$ it ends inside the arc of radius $\|\boldsymbol{y}\|$ . Multiplying $\dfrac{|\boldsymbol{x}\cdot \boldsymbol{y}|}{\|\boldsymbol{x}\|}\le\|\boldsymbol{y}\|$ by $\|\boldsymbol{x}\|$ gives the inequality; equality holds when $\boldsymbol{y}$ already lies on the line of $\boldsymbol{x}$ , so that the segment reaches the arc.

Drag $\boldsymbol{x}$ and $\boldsymbol{y}$ : $|\boldsymbol{x}\cdot \boldsymbol{y}|$ never exceeds $\|\boldsymbol{x}\|\|\boldsymbol{y}\|$ , because the shadow of $\boldsymbol{x}$ on the line of $\boldsymbol{y}$ is never longer than $\boldsymbol{x}$ . Equality holds exactly when the vectors lie on one line.

The inequality holds for all vectors, including $\boldsymbol{0}$ , where both sides are $0$ . Equality holds if and only if one vector is a multiple of the other, that is, when both lie on one line through the origin. The absolute value matters: $\boldsymbol{x}\cdot(-\boldsymbol{x})=-\|\boldsymbol{x}\|^2$ is very negative for long $\boldsymbol{x}$ ; only the size of the inner product is bounded.

The inequality is what makes the angle between vectors well defined in $\mathbb{R}^n$ , since a cosine must lie in $[-1,1]$ , and it is the key step in proving the triangle inequality $\|\boldsymbol{x}+\boldsymbol{y}\|\le\|\boldsymbol{x}\|+\|\boldsymbol{y}\|$ $∥ y ∥$ . Its proof uses only the defining properties of an inner product, so it holds for every inner product with $\|\boldsymbol{x}\|=\sqrt{\langle\boldsymbol{x},\boldsymbol{x}\rangle}$ $⟨ x, x ⟩$ . It returns later in a version for random variables.

Key Results¶

Cauchy-Schwarz justifies the formula for the angle between vectors.
It can be used to prove the triangle inequality.

Examples¶

For orthogonal vectors, the left side is $0$ .
For $\boldsymbol{x}=\boldsymbol{y}$ $y$ , equality holds.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 5

Aliases: Cauchy-Bunyakovsky-Schwarz inequality, Schwarz inequality, Cauchy's inequality, CBS inequality, Cauchy-Schwarz Symbols: real numbers, square root

Last updated: 20 October 2026 03:02