# spantwo

sum and intersection of subspaces

## Calling Sequence

    [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

- spanplus
- spaninter