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spantwo

sum and intersection of subspaces

Syntax

[Xp,dima,dimb,dim]=spantwo(A,B, [tol])

Arguments

A, B

two real or complex matrices with equal number of rows

Xp

square non-singular matrix

dima, dimb, dim

integers, dimension of subspaces

tol

nonnegative real number

Description

Given two matrices A and B with same number of rows, returns a square matrix Xp (non singular but not necessarily orthogonal) such that :

[A1, 0]    (dim-dimb rows)
Xp*[A,B]=[A2,B2]    (dima+dimb-dim rows)
[0, B3]    (dim-dima rows)
[0 , 0]

The first dima columns of inv(Xp) span range(A).

Columns dim-dimb+1 to dima of inv(Xp) span the intersection of range(A) and range(B).

The dim first columns of inv(Xp) span range(A)+range(B).

Columns dim-dimb+1 to dim of inv(Xp) span range(B).

Matrix [A1;A2] has full row rank (=rank(A)). Matrix [B2;B3] has full row rank (=rank(B)). Matrix [A2,B2] has full row rank (=rank(A inter B)). Matrix [A1,0;A2,B2;0,B3] has full row rank (=rank(A+B)).

Examples

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));

See also


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