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Cauchy–Schwarz: , with equality exactly for linearly dependent vectors
Statement
For vectors in a real or complex inner product space,
Equality holds if and only if and are linearly dependent, including the case in which either vector is zero.
Facts & Assumptions
Given: Vectors in an inner product space over or .
The inner product is linear in the first variable, conjugate-linear in the second, conjugate symmetric, and positive definite (Real and complex inner product spaces, with the inner product linear in the first argument).
The norm is the nonnegative square root of the diagonal pairing (The norm induced by a real or complex inner product, Existence and uniqueness of -th roots: a unique with ).
Complex modulus satisfies and vanishes exactly at zero (Conjugation is an involutive real-field automorphism, , and modulus is definite, multiplicative, and subadditive).
A two-vector list is dependent exactly when a nontrivial scalar combination vanishes (Linear independence: a finite list is independent when forces every , and a subset is independent when every injective finite list into is independent).
Proof
If , both sides are zero and the pair is dependent. Suppose , put , and use [L1] to expand .
Conversely, if are dependent and neither is zero, write ; then . If either is zero, equality is immediate.
Multiplying step 1.1 by the positive number gives . Since both sides of the desired inequality are nonnegative, factoring the difference of their squares gives the stated inequality.
Under , equality in step 2.1 holds exactly when , which by [L1] is exactly . Thus equality implies dependence. The already separated case does too.
Steps 2.1, 3.1, and 1.2 prove the inequality and both equality directions.
Depends on
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product
- Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
- Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$
- Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i<n} \lambda_i v_i = 0_V$ forces every $\lambda_i = 0_F$, and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent
Used by
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality Corollary
- Pointwise norm and angle from a riemannian metric Definition
- Norm-attaining functionals on a Hilbert space Example
- The Gram determinant of two vectors is ‖ u‖²‖ v‖²-|⟨ u,v⟩|² and detects dependence Example
- Cheeger sweep and layer cake Lemma
- Chevalley basis and real structure constants Lemma
- Expander walk contraction Lemma
- Finite torus fourier orthogonality and affine change Lemma
- Overlap control gives a union lower bound Lemma
- Violated-edge positions have controlled collisions Lemma
- Expander walk hits bad edges Proposition
- On ℝⁿ, abstract Cauchy–Schwarz is exactly the published finite-sum Cauchy–Schwarz inequality Proposition
- Expander mixing lemma Theorem
- For a unit vector x, the smallest perturbation making (μ,x) an exact eigenpair has spectral norm ‖Ax-μ x‖₂ Theorem
- Hilbert spaces are reflexive by Riesz representation Theorem
- Near a simple Hermitian eigenvector, Rayleigh-quotient iteration converges cubically Theorem
- Rank-two root-system classification Theorem
Dependency tree · two levels
35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §6A (standard reference, not scraped)
- Sergei Treil, Linear Algebra Done Wrong, Ch. 5, §5.1 (standard reference, not scraped)