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Scilab help >> Linear Algebra > spantwo

# 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));```

F. D.

### Comments

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