how to obtain the basis vector of the space orthogonal to K vectors?

Hi,

I have Psi= [e1; e2; ... eK; ] having K (eigen vectors) rows, and need to find (N-K) basis vectors of the space orthogonal to K vectors. So that resultant, Psi_new= [e1; e2; ... eK; e(K+1); ... eN; ].

If K=4 and N=6. Then, for any e=[0.9999; 0.8294; 1.0173; 0.9973]; Psi=[e']; Further, I have to find (6-4=2) basis vectors.

basis=null(Psi(:).') result as

basis =

-0.4302 0.8781 -0.1495 -0.1465

-0.5276 -0.1495 0.8167 -0.1797

-0.5173 -0.1465 -0.1797 0.8238

which is not the required one. So, could anyone tell me what commands in matlab is useful to obtain the (N-K) basis vector of the space orthogonal to K vectors?

Answer 1

[John D'Errico]

null is the tool to do exactly that.

[Syam MS]

So does that mean I just need to choose 2 elements from the 3x4 basis matrix ?

[John D'Errico]

NO. You need to use the function null. Read the help for null. Look at the examples. TRY IT.

For example, Consider the vectors in B.

B = rand(3,5)
B =
       0.8892       0.7917      0.46752      0.91457       0.3652
      0.66566      0.90643      0.91953      0.93016      0.78042
      0.96681     0.068954      0.96643      0.86964      0.22151

They can be viewed as independent vectors spanning some 3-dimensional subspace of R^5. While they are not orthogonal, they could eaily be made so, using for example, a Gram-Schmidt orthogonalization. Or you could use just orth.

Since here we have a 5 dimensional space, and 3 vectors spanning some 3 dimensional subspace, then there is a subspace of dimension 5-2 that is orthogonal to this subspace. null provides us a basis for the nullspace.

N = null(B)
N =
     -0.53286      0.25093
     -0.19447     -0.43255
     -0.15403     -0.44708
      0.80335     0.063967
     -0.09563      0.73889
>> B*N
ans =
  -9.0206e-17   1.1102e-16
   1.9429e-16  -1.1102e-16
   9.3675e-17   1.6653e-16

As you see, to within floating point trash, they are orthogonal to the original set, and they form an orthogonal basis, again, to within floating point trash.

N'*N
ans =
            1  -6.8656e-18
  -6.8656e-18            1

Read the help for null. Look at the examples.