Abstract Linear Algebra 12 | Cauchy-Schwarz Inequality [dark version]

TL;DR

An inner product induces a norm via ||X|| = √⟨X, X⟩, turning algebraic vector spaces into geometric ones.

Briefing Cornell Notes

Briefing

Cauchy–Schwarz inequality is the engine behind how inner products turn algebraic vector spaces into geometric ones: it bounds the inner product of two vectors by their lengths, and that bound is what makes “angles” and “projections” mathematically precise. In any real or complex inner product space, the inequality says that the absolute value of ⟨X, Y⟩ never exceeds ||X||·||Y||. Equality is tightly constrained: it happens only when X and Y are linearly dependent, meaning the vectors lie on the same line—so the inequality becomes a sharp test for when two directions align.

The path to that result starts with the basic payoff of having an inner product. Once an inner product ⟨·,·⟩ is available, the space gains a notion of length via the norm ||X|| = √⟨X, X⟩. This works because the inner product’s positive-definite property guarantees ⟨X, X⟩ ≥ 0, with equality only for the zero vector. With lengths in hand, the inequality can be interpreted as a universal constraint on how large the “overlap” ⟨X, Y⟩ can be relative to the sizes of X and Y.

The proof uses a normalization trick and a quadratic-polynomial argument. First, the trivial case X = 0 makes ⟨X, Y⟩ = 0 immediately, and the inequality holds since the right-hand side is also zero. For the nonzero case, the proof divides by ||X|| and reduces the problem to vectors where ||X|| = 1. Fix a real scalar Λ and consider the nonnegative quantity ⟨Y − ΛX, Y − ΛX⟩. Expanding with bilinearity (and accounting for conjugate linearity in the first slot, though the real choice of Λ keeps the algebra clean) yields a quadratic expression in Λ:

⟨Y − ΛX, Y − ΛX⟩ = ||Y||² − 2Λ·Re⟨Y, X⟩ + Λ².

Because ⟨Y − ΛX, Y − ΛX⟩ must be ≥ 0 for every real Λ, the quadratic polynomial cannot dip below zero. That forces the discriminant to be nonpositive, which in turn implies (Re⟨Y, X⟩)² ≤ ||Y||² when ||X|| = 1. Taking square roots gives |Re⟨Y, X⟩| ≤ ||Y||. For real vector spaces, the inner product itself is real, so this becomes the full Cauchy–Schwarz bound.

For complex vector spaces, the proof finishes by “rotating” the complex number ⟨X, Y⟩ onto the real axis. Multiplying by a unit-modulus complex scalar (a phase factor) turns the inner product into one whose real part equals its absolute value. Applying the already-proved real case to the rotated setup recovers the general inequality |⟨X, Y⟩| ≤ ||X||·||Y||. The result is universal, reusable across mathematics, and it clarifies exactly when equality occurs—precisely when the vectors are linearly dependent—linking the algebra of inner products to the geometry of angles and distances.

Cornell Notes

Cauchy–Schwarz inequality is the key rule that connects inner products to geometry. In any real or complex inner product space, it guarantees that |⟨X, Y⟩| ≤ ||X||·||Y||, where ||X|| = √⟨X, X⟩. The proof starts by handling X = 0, then normalizes to the case ||X|| = 1. For real spaces, it considers ⟨Y − ΛX, Y − ΛX⟩ ≥ 0 for all real Λ, expands it into a quadratic in Λ, and uses the fact that a quadratic nonnegative for all Λ must have nonpositive discriminant. For complex spaces, it rotates ⟨X, Y⟩ by a unit complex factor so its real part becomes the absolute value, reducing the complex case to the real one.

Why does having an inner product automatically give a notion of length in a vector space?

An inner product ⟨X, X⟩ is always nonnegative because of positive definiteness, and it equals 0 only when X is the zero vector. Defining the norm by ||X|| = √⟨X, X⟩ turns that nonnegative quantity into a length scale for every vector, enabling geometric interpretations like distances and angle-related bounds.

How does the proof reduce Cauchy–Schwarz to the case ||X|| = 1?

For nonzero X, the argument divides by ||X|| so that the normalized vector X̂ = X/||X|| has norm 1. Since the inequality is homogeneous in X and Y, proving |⟨Y, X̂⟩| ≤ ||Y|| (with ||X̂|| = 1) is enough to recover the general bound |⟨X, Y⟩| ≤ ||X||·||Y||.

What nonnegative expression is expanded, and why must it stay nonnegative?

The proof fixes a real scalar Λ and expands ⟨Y − ΛX, Y − ΛX⟩. Positive definiteness of the inner product implies ⟨v, v⟩ ≥ 0 for every vector v, so ⟨Y − ΛX, Y − ΛX⟩ ≥ 0 for all real Λ. That constraint on the expanded form drives the inequality.

How does the quadratic-in-Λ argument force a bound on Re⟨Y, X⟩?

After normalization ||X|| = 1, the expansion becomes ||Y||² − 2Λ·Re⟨Y, X⟩ + Λ². This is a quadratic polynomial in Λ that must be ≥ 0 for every real Λ. A quadratic that is never negative cannot have two distinct real roots, so its discriminant must be ≤ 0, yielding (Re⟨Y, X⟩)² ≤ ||Y||² and thus |Re⟨Y, X⟩| ≤ ||Y||.

How does the complex case get reduced to the real one?

In complex spaces, ⟨X, Y⟩ can be complex. The proof multiplies by a unit-modulus complex scalar (a phase factor) so that the rotated inner product lies on the real axis. Because both sides of the inequality are real after taking real parts, applying the real-space result to the rotated setup produces |⟨X, Y⟩| ≤ ||X||·||Y||.

Review Questions

In an inner product space, which property ensures that ||X|| = √⟨X, X⟩ is well-defined and equals 0 only for X = 0?
Why does requiring ⟨Y − ΛX, Y − ΛX⟩ ≥ 0 for all real Λ imply a discriminant condition for the quadratic in Λ?
What role does multiplying by a unit-modulus complex number play in extending Cauchy–Schwarz from real to complex vector spaces?

Key Points

1
An inner product induces a norm via ||X|| = √⟨X, X⟩, turning algebraic vector spaces into geometric ones.
2
Cauchy–Schwarz inequality states that for any vectors X and Y, |⟨X, Y⟩| ≤ ||X||·||Y|| in real and complex inner product spaces.
3
Equality in Cauchy–Schwarz occurs only when X and Y are linearly dependent, i.e., they lie on the same line.
4
The proof uses the nonnegativity of ⟨Y − ΛX, Y − ΛX⟩ for all real Λ, which follows from positive definiteness.
5
Normalizing to ||X|| = 1 simplifies the expansion to a quadratic polynomial in Λ.
6
For complex spaces, a unit-modulus phase factor rotates ⟨X, Y⟩ so its real part matches its absolute value, reducing the complex case to the real argument.

Highlights

Cauchy–Schwarz turns “angle” intuition into a theorem by bounding inner products using only lengths: |⟨X, Y⟩| ≤ ||X||·||Y||.

The proof hinges on the fact that ⟨Y − ΛX, Y − ΛX⟩ is always ≥ 0, forcing a discriminant constraint.

After normalization ||X|| = 1, the expansion becomes ||Y||² − 2Λ·Re⟨Y, X⟩ + Λ², and nonnegativity for all Λ yields (Re⟨Y, X⟩)² ≤ ||Y||².

Complex vector spaces are handled by rotating the inner product with a unit-modulus complex scalar so the real-part argument applies to the absolute value.

Topics

Inner Products
Norms
Cauchy-Schwarz Inequality
Quadratic Discriminant
Complex Phase Rotation