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));
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