DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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The quadratic energy $ϕ (x) = \frac{1}{2} ⟨ A x, x ⟩ - Re ⟨ b, x ⟩$ of a Hermitian positive-definite linear system

Definition

Let $A \in M_{n} (C)$ be Hermitian positive definite, and let $b \in C^{n}$ . The quadratic energy of the linear system $A x = b$ is the real-valued function

$ϕ : C^{n} \to R, ϕ (x) : = \frac{1}{2} ⟨ A x, x ⟩ - Re ⟨ b, x ⟩,$

where the coordinate inner product is the one from The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products. The diagonal term is real because $⟨ A x, x ⟩ = ∥ x ∥_{A}^{2}$ by The energy inner product $⟨ u, v ⟩_{A} = ⟨ A u, v ⟩$ and energy norm $∥ u ∥_{A}$ for a Hermitian positive-definite matrix.

Depends on

Used by

For Hermitian positive-definite A, the exact solution of Ax=b is the unique minimizer of the quadratic energy Theorem

Dependency tree · two levels

7 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

Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (standard reference, not scraped)

The quadratic energy ϕ(x)=12⟨Ax,x⟩−Re⁡⟨b,x⟩ϕ(x)=21​⟨Ax,x⟩−Re⟨b,x⟩ of a Hermitian positive-definite linear system