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<div id="version" align=right><b>petsc-3.7.5 2017-01-01</b></div>
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<A NAME="MatSetTransposeNullSpace"><H1>MatSetTransposeNullSpace</H1></A>
attaches a null space to a matrix.
<H3><FONT COLOR="#CC3333">Synopsis</FONT></H3>
<PRE>
#include "petscmat.h"
PetscErrorCode MatSetTransposeNullSpace(Mat mat,MatNullSpace nullsp)
</PRE>
Logically Collective on <A HREF="../Mat/Mat.html#Mat">Mat</A> and <A HREF="../Mat/MatNullSpace.html#MatNullSpace">MatNullSpace</A>
<P>
<H3><FONT COLOR="#CC3333">Input Parameters</FONT></H3>
<TABLE border="0" cellpadding="0" cellspacing="0">
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>mat </B></TD><TD>- the matrix
</TD></TR>
<TR><TD WIDTH=40></TD><TD ALIGN=LEFT VALIGN=TOP><B>nullsp </B></TD><TD>- the null space object
</TD></TR></TABLE>
<P>
<P>
<H3><FONT COLOR="#CC3333">Notes</FONT></H3>
For inconsistent singular systems (linear systems where the right hand side is not in the range of the operator) this allows the linear system to be solved in a least squares sense.
You must also call <A HREF="../Mat/MatSetNullSpace.html#MatSetNullSpace">MatSetNullSpace</A>()
<P>
<P>
The fundamental theorem of linear algebra (Gilbert Strang, Introduction to Applied Mathematics, page 72) states that
the domain of a matrix A (from R^n to R^m (m rows, n columns) R^n = the direct sum of the null space of A, n(A), + the range of A^T, R(A^T).
Similarly R^m = direct sum n(A^T) + R(A). Hence the linear system A x = b has a solution only if b in R(A) (or correspondingly b is orthogonal to
n(A^T)) and if x is a solution then x + alpha n(A) is a solution for any alpha. The minimum norm solution is orthogonal to n(A). For problems without a solution
the solution that minimizes the norm of the residual (the least squares solution) can be obtained by solving A x = \hat{b} where \hat{b} is b orthogonalized to the n(A^T).
<P>
Krylov solvers can produce the minimal norm solution to the least squares problem by utilizing <A HREF="../Mat/MatNullSpaceRemove.html#MatNullSpaceRemove">MatNullSpaceRemove</A>().
<P>
<P>
<H3><FONT COLOR="#CC3333">See Also</FONT></H3>
<A HREF="../Mat/MatCreate.html#MatCreate">MatCreate</A>(), <A HREF="../Mat/MatNullSpaceCreate.html#MatNullSpaceCreate">MatNullSpaceCreate</A>(), <A HREF="../Mat/MatSetNearNullSpace.html#MatSetNearNullSpace">MatSetNearNullSpace</A>(), <A HREF="../Mat/MatGetNullSpace.html#MatGetNullSpace">MatGetNullSpace</A>(), <A HREF="../Mat/MatSetNullSpace.html#MatSetNullSpace">MatSetNullSpace</A>(), <A HREF="../Mat/MatGetTransposeNullSpace.html#MatGetTransposeNullSpace">MatGetTransposeNullSpace</A>(), <A HREF="../Mat/MatNullSpaceRemove.html#MatNullSpaceRemove">MatNullSpaceRemove</A>()
<BR><P><B><P><B><FONT COLOR="#CC3333">Level:</FONT></B>advanced
<BR><FONT COLOR="#CC3333">Location:</FONT></B><A HREF="../../../src/mat/interface/matrix.c.html#MatSetTransposeNullSpace">src/mat/interface/matrix.c</A>
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