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

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The energy inner product $⟨ u, v ⟩_{A} = ⟨ A u, v ⟩$ and energy norm $∥ u ∥_{A}$ for a Hermitian positive-definite matrix

Definition

Let $A \in M_{n} (C)$ be Hermitian positive definite in the sense of Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal. For vectors $u, v \in C^{n}$ , define the energy inner product

$⟨ u, v ⟩_{A} : = ⟨ A u, v ⟩,$

where $⟨ \cdot, \cdot ⟩$ is the standard coordinate inner product of 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. Because $A$ is Hermitian positive definite, this is an inner product on $C^{n}$ , and its induced norm

$∥ u ∥_{A} : = \sqrt{⟨ u, u ⟩_{A}}$

is the energy norm of $u$ .

Equivalently, if $A = L L^{*}$ is the Cholesky factorization from Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal, then $⟨ u, v ⟩_{A} = ⟨ L^{*} u, L^{*} v ⟩$ and $∥ u ∥_{A} = ∥ L^{*} u ∥_{2}$ .

Depends on

Used by

Dependency tree · two levels

8 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

Magnus R. Hestenes and Eduard Stiefel, Methods of Conjugate Gradients for Solving Linear Systems (standard reference, not scraped)
Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain (standard reference, not scraped)

The energy inner product ⟨u,v⟩A=⟨Au,v⟩⟨u,v⟩A​=⟨Au,v⟩ and energy norm ∥u∥A∥u∥A​ for a Hermitian positive-definite matrix