Adjoint equation: Difference between revisions

Revision as of 16:17, 7 November 2023

An equation of the form x₀ = 𝗟^Tx₁, in which the linear operator 𝗟^T is the adjoint of the matrix operator 𝗟 that satisfies (𝗟^Tx₁,x₀) = (x₁,𝗟x₀), where x₀ and x₁ are vectors and (,) represents an inner product.
If (,) is the standard dot product (Euclidean inner product) then 𝗟^T is simply the transpose of 𝗟.
See adjoint sensitivity, adjoint model, tangent linear equation.

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+{{Term
+|Display title=adjoint equation
-{{TermHeader}}
+|Definitions={{Definition
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+|Num=1
+|Meaning=
-<div class="termentry">
+An equation of the form '''x'''<sub>0</sub> = &#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>, in which the [[linear operator]] &#x1D5DF;<sup>''T''</sup> is the adjoint  of the matrix [[operator]] &#x1D5DF; that satisfies (&#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>,'''x'''<sub>0</sub>) = ('''x'''<sub>1</sub>,&#x1D5DF;'''x'''<sub>0</sub>), where '''x'''<sub>0</sub> and '''x'''<sub>1</sub> are vectors and (,)  represents an [[inner product]].<br/> If (,) is the standard dot product (Euclidean inner product) then &#x1D5DF;<sup>''T''</sup> is simply the transpose of  &#x1D5DF;. <br/>''See'' [[adjoint sensitivity]], [[adjoint model]], [[tangent linear equation]].
-  <div class="term">
+}}
-== adjoint equation ==
+}}
-  </div>
-<div class="definition"><div class="short_definition">An equation of the form '''x'''<sub>0</sub> = &#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>, in which the [[linear operator]] &#x1D5DF;<sup>''T''</sup> is the adjoint  of the matrix [[operator]] &#x1D5DF; that satisfies (&#x1D5DF;<sup>''T''</sup>'''x'''<sub>1</sub>,'''x'''<sub>0</sub>) = ('''x'''<sub>1</sub>,&#x1D5DF;'''x'''<sub>0</sub>), where '''x'''<sub>0</sub> and '''x'''<sub>1</sub> are vectors and (,)  represents an [[inner product]].</div><br/> <div class="paragraph">If (,) is the standard dot product (Euclidean inner product) then &#x1D5DF;<sup>''T''</sup> is simply the transpose of  &#x1D5DF;. <br/>''See'' [[adjoint sensitivity]], [[adjoint model]], [[tangent linear equation]].</div><br/> </div>
-</div>
-{{TermIndex}}
-{{TermFooter}}
-[[Category:Terms_A]]