Adjoint equation: Difference between revisions

Latest revision as of 21:59, 13 January 2024

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.

INDEX
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
0
1
2
3
4
5
6
7
8
9

@@ Line 3: / Line 3: @@
 |Definitions={{Definition
 |Num=1
-|Meaning=
+|Meaning=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]].
-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]].
+|Explanation=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]].
 }}
 }}