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The standard formulas on and on are inner products
Statement
For and , the formulas
define inner products, linear in the first argument. At , the unique pairing on the zero space is an inner product.
Facts & Assumptions
Given: A natural number and the two displayed coordinate pairings.
Finite products in a commutative monoid have an empty value and may be read additively as finite sums (The product of a finite list in a monoid, by recursion, with the empty product () equal to the identity).
Complex conjugation preserves sums and products, and , with equality exactly when (Conjugation is an involutive real-field automorphism, , and modulus is definite, multiplicative, and subadditive).
The standard unit vectors form an ordered basis of , including (The standard list with and for is an ordered basis of ; hence , and is the zero space with basis and dimension ).
An inner product is linear in the first argument, conjugate symmetric, positive on the diagonal, and definite (Real and complex inner product spaces, with the inner product linear in the first argument).
The reals form a totally ordered field, so addition preserves order; the square of a nonzero real number is positive (The reals form a totally ordered field, Squares of nonzero elements are positive).
Proof
Distributivity of the finite sums in [L1] gives linearity in the first variable. In the complex case [L2] gives conjugate-linearity in the second and conjugate symmetry; in the real case conjugation is the identity.
For a finite list of nonnegative real numbers , put and as in [L1]. By [L5], if and , then and . Hence holds exactly when both and . Starting with the empty sum, induction proves and exactly when every . On the diagonal, take in the real formula and in the complex formula. By [L5], is nonnegative and equals zero exactly when ; [L2] gives the corresponding assertion for . Thus both diagonal pairings are real and nonnegative, and each vanishes exactly when all coordinates vanish, which by [L3] means the vector is zero.
Hence all the axioms in [L4] hold. When , the sum is empty and equals , while the zero vector is the only vector, so definiteness is valid.
Depends on
- Real and complex inner product spaces, with the inner product linear in the first argument
- The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity
- The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$
- Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
- The reals form a totally ordered field
- Squares of nonzero elements are positive
Used by
- Certified-action implementation of exponential Euler and ETD2 Algorithm
- Finite locally linear embedding with minimum-norm weights and a simple constant kernel Algorithm
- Flexible GMRES with variable right preconditioning Algorithm
- Steady MINI Stokes pressure-mass Schur bounds Corollary
- Discrete linear input-state-output experiments Definition
- Finite registers and their ordered computational bases Definition
- Galerkin coarse matrices and the exact coarse-correction error map Definition
- Ordered tensor registers, their inner product, and wire permutations Definition
- The energy inner product ⟨ u,v⟩_A=⟨ Au,v⟩ and energy norm ‖u‖_A for a Hermitian positive-definite matrix Definition
- The Euclidean logarithmic norm of a complex matrix Definition
- The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
- The phi-function family by everywhere-convergent series Definition
- The quadratic energy φ(x)=1/2⟨ Ax,x⟩-Re⟨ b,x⟩ of a Hermitian positive-definite linear system Definition
- A diagonal stiff diffusion step with linear reaction Example
- A finite-matrix Schrödinger split preserves norm Example
- A projective measurement branch with probability zero Example
- A singular matrix has a polar decomposition with visibly nonunique isometric factors Example
- A worked polar decomposition of an invertible matrix Example
- A worked QR factorisation of an invertible real matrix Example
- An explicit Hermitian 2x2 matrix is unitarily diagonalised Example
- An explicit real symmetric 3x3 matrix is orthogonally diagonalised Example
- Bell-state Schmidt decomposition and reduced density Example
- Bessel's inequality is strict for a vector outside the span of a proper orthonormal set Example
- Courant-Fischer is checked on an explicit 3x3 symmetric matrix Example
- Dirac and rank-one coordinate check Example
- Exact exponential stepping captures nonnormal transient growth Example
- Forward Euler creates an observation-map error Example
- Gram–Schmidt on an explicit basis of ℂ² with conjugation visible Example
- Gram–Schmidt on an explicit basis of ℝ³ Example
- Projection onto a plane in ℝ³ and the nearest-point calculation Example
- Qubit dephasing channel in Kraus and Choi forms Example
- Real-time unitary and imaginary-time finite-matrix propagation Example
- The adjoint of an explicit map ℂ²→ℂ³ is its conjugate-transpose matrix Example
- The eigenvalues of a principal 2x2 submatrix interlace those of a 3x3 symmetric matrix Example
- The non-negative square root of an explicit matrix is exhibited as a polynomial in the matrix Example
- The real quarter-turn is normal and appears as a single 2x2 block in the real normal classification Example
- The Riesz representative of an explicit functional on ℂ³ Example
- The singular value decomposition of an explicit 2x3 matrix Example
- Unequal replication removes factorial-column orthogonality Example
- FALSE: Every complex inner product satisfies ⟨ u,v⟩=⟨ v,u⟩ False statement
…and 42 more results.
Dependency tree · two levels
40 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, Example 6.3 (standard reference, not scraped)