The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products

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

Used by

• The energy inner product ⟨ u,v⟩_A=⟨ Au,v⟩ and energy norm ‖u‖_A for a Hermitian positive-definite matrix Definition
• The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices Definition
• The quadratic energy φ(x)=1/2⟨ Ax,x⟩-Re⟨ b,x⟩ of a Hermitian positive-definite linear system Definition
• 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
• 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
• 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
• 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
• FALSE: Every complex inner product satisfies ⟨ u,v⟩=⟨ v,u⟩ False statement
• FALSE: Every orthogonal set is an orthonormal basis False statement
• FALSE: On a complex inner product space, (λ T)^*=λ T^* for every scalar False statement
• If v₁,…,vₘ∈ℝⁿ satisfy ⟨ vᵢ,vⱼ⟩=t≥0 for i≠ j and ⟨ vᵢ,vᵢ⟩>t, they are linearly independent Lemma
• Compact Lie groups admit bi-invariant metrics Proposition
• On ℝⁿ, abstract Cauchy–Schwarz is exactly the published finite-sum Cauchy–Schwarz inequality Proposition
• A real normal endomorphism is orthogonally block-diagonalisable with 1x1 real blocks and 2x2 rotation-scaling blocks Theorem
• Finite-dimensional compact-group representations are unitarizable Theorem

Dependency tree · two levels

31 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)