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Untitled[edit]

I might be mistaken, but I didn't think that Polyharmonic splines actually do guarantee that the linear system matrix is positive definite, just that it's nonsingular. For example, consider phi(r) = r with centers 0 and 1; the matrix is

$M\,=\,{\begin{bmatrix}0&&1&&1&&0\\1&&0&&1&&1\\1&&1&&0&&0\\0&&1&&0&&0\end{bmatrix}}$

which is not positive definite:

${\begin{bmatrix}1&&0&&-1&&0\end{bmatrix}}M{\begin{bmatrix}1\\0\\-1\\0\end{bmatrix}}=-2$

Can somebody with more experience than I verify this?

128.143.137.224 (talk) 18:10, 1 October 2009 (UTC)[reply]

Thats why there is a polynomial, namely to avoid the PSD matrices. See "Spline Models for Observational Data" by Wahba, Page 31. — Preceding unsigned comment added by Hannes36743 (talk • contribs) 17:13, 3 December 2013 (UTC)[reply]

What's T?[edit]

In the definition section, which is otherwise quite good, what is "T"? As in, the term superscripted on so many of the matrices....? — Preceding unsigned comment added by 50.26.246.186 (talk) 17:02, 19 February 2016 (UTC)[reply]

Matrix transpose, this is now explicitly stated in definition section Jrheller1 (talk) 19:02, 20 February 2016 (UTC)[reply]

O.K. Maybe it threw me off because it seems strange to declare B as a transpose, then to also transpose it in the constraint equation. In any case, definitely a good addition to the text! — Preceding unsigned comment added by 50.26.246.186 (talk) 22:55, 22 February 2016 (UTC)[reply]

That's a common notation in Mathematics. It saves a lot of space if the matrix is long and thin. Column vectors are also usually declared as the transpose of a row vector for the same reason. 93.132.186.56 (talk) 10:42, 9 March 2023 (UTC)[reply]

Clarification on additional constraints[edit]

In the section 'additional constraints' two systems of linear equations are derived:

 $A(A\mathbf {w} +B\mathbf {v} -\mathbf {f} +\lambda C\mathbf {w} )=0$  and  $B^{\textrm {T}}(A\mathbf {w} +B\mathbf {v} -\mathbf {f} )=0.$

Then it is stated that $A$ is invertible. Is this actually true? Why?

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 ==Untitled==
 I might be mistaken, but I didn't think that Polyharmonic splines actually do guarantee that the linear system matrix is positive definite, just that it's nonsingular. For example, consider phi(r) = r with centers 0 and 1; the matrix is
@@ Line 21: / Line 20: @@
 Thats why there is a polynomial, namely to avoid the PSD matrices. See "Spline Models for Observational Data" by Wahba, Page 31.  <small><span class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Hannes36743|Hannes36743]] ([[User talk:Hannes36743|talk]] • [[Special:Contributions/Hannes36743|contribs]]) 17:13, 3 December 2013 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
+==What's T?==
+In the definition section, which is otherwise quite good, what is "T"?  As in, the term superscripted on so many of the matrices....?  <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/50.26.246.186|50.26.246.186]] ([[User talk:50.26.246.186|talk]]) 17:02, 19 February 2016 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
+Matrix transpose, this is now explicitly stated in definition section [[User:Jrheller1|Jrheller1]] ([[User talk:Jrheller1|talk]]) 19:02, 20 February 2016 (UTC)
+O.K.  Maybe it threw me off because it seems strange to declare B as a transpose, then to also transpose it in the constraint equation.  In any case, definitely a good addition to the text!  <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/50.26.246.186|50.26.246.186]] ([[User talk:50.26.246.186|talk]]) 22:55, 22 February 2016 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
+:That's a common notation in Mathematics. It saves a lot of space if the matrix is long and thin. Column vectors are also usually declared as the transpose of a row vector for the same reason. [[Special:Contributions/93.132.186.56|93.132.186.56]] ([[User talk:93.132.186.56|talk]]) 10:42, 9 March 2023 (UTC)
+== Clarification on additional constraints ==
+In the section 'additional constraints' two systems of linear equations are derived:
+ <math> A(A\mathbf{w} + B\mathbf{v} - \mathbf{f} +\lambda C \mathbf{w}) = 0 </math> and <math> B^{\textrm{T}}(A\mathbf{w} + B\mathbf{v} - \mathbf{f}) = 0.</math>
+Then it is stated that <math>A</math> is invertible. Is this actually true? Why?