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.TH spantwo 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
spantwo - sum and intersection of subspaces
.SH CALLING SEQUENCE
.nf
[Xp,dima,dimb,dim]=spantwo(A,B, [tol])
.fi
.SH PARAMETERS
.TP 10
A, B
: two real or complex matrices with equal number of rows
.TP
Xp
: square non-singular matrix
.TP
dima, dimb, dim
: integers, dimension of subspaces
.TP
tol
: nonnegative real number
.SH DESCRIPTION
Given two matrices \fVA\fR and \fVB\fR with same number of rows,
returns a square matrix \fVXp\fR (non singular but not necessarily orthogonal)
such that :
.LP
.nf
[A1, 0] (dim-dimb rows)
Xp*[A,B]=[A2,B2] (dima+dimb-dim rows)
[0, B3] (dim-dima rows)
[0 , 0]
.fi
The first \fVdima\fR columns of \fVinv(Xp)\fR span range(\fVA\fR).
.LP
Columns \fVdim-dimb+1\fR to \fVdima\fR of \fVinv(Xp)\fR span the
intersection of range(A) and range(B).
.LP
The \fVdim\fR first columns of \fVinv(Xp)\fR span
range(\fVA\fR)+range(\fVB\fR).
.LP
Columns \fVdim-dimb+1\fR to \fVdim\fR of \fVinv(Xp)\fR span
range(\fVB\fR).
.LP
Matrix \fV[A1;A2]\fR has full row rank (=rank(A)). Matrix \fV[B2;B3]\fR has
full row rank (=rank(B)). Matrix \fV[A2,B2]\fR has full row rank (=rank(A inter B)). Matrix \fV[A1,0;A2,B2;0,B3]\fR has full row rank (=rank(A+B)).
.SH EXAMPLE
.nf
A=[1,0,0,4;
5,6,7,8;
0,0,11,12;
0,0,0,16];
B=[1,2,0,0]';C=[4,0,0,1];
Sl=ss2ss(syslin('c',A,B,C),rand(A));
[no,X]=contr(Sl('A'),Sl('B'));CO=X(:,1:no); //Controllable part
[uo,Y]=unobs(Sl('A'),Sl('C'));UO=Y(:,1:uo); //Unobservable part
[Xp,dimc,dimu,dim]=spantwo(CO,UO); //Kalman decomposition
Slcan=ss2ss(Sl,inv(Xp));
.fi
.SH SEE ALSO
spanplus, spaninter
.SH AUTHOR
F. D.
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