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Difference between revisions of "Transposed equations"

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The transposed equations of a linear algebraic system are the linear equations with the [[Transposed matrix|transposed matrix]] as coefficient matrix.
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In the case of linear integral equations (cf. [[Integral equation|Integral equation]]), transposed equations are equations with kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093940/t0939401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093940/t0939402.png" />.
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The transposed equations of a linear algebraic system are the linear equations with the [[transposed matrix]] as coefficient matrix.
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In the case of linear integral equations (cf. [[Integral equation]]), transposed equations are equations with kernels $K(x,y)$ and $K(y,x)$.
  
 
Transposed equations are also called adjoint equations or associated equations.
 
Transposed equations are also called adjoint equations or associated equations.
  
 
====Comments====
 
====Comments====
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The transposed equation of an integral equation plays an important role in the [[Fredholm alternative|Fredholm alternative]].
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The transposed equation of an integral equation plays an important role in the [[Fredholm alternative]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR>
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</table>

Latest revision as of 20:32, 21 November 2014


The transposed equations of a linear algebraic system are the linear equations with the transposed matrix as coefficient matrix.

In the case of linear integral equations (cf. Integral equation), transposed equations are equations with kernels $K(x,y)$ and $K(y,x)$.

Transposed equations are also called adjoint equations or associated equations.

Comments

The transposed equation of an integral equation plays an important role in the Fredholm alternative.

References

[a1] K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)
[a2] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
How to Cite This Entry:
Transposed equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transposed_equations&oldid=34732
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