Sparse Matrix inversion some time singular some time get a big value

Question

I want to invert a matrix which is a "band" diagonal matrix. The structure of the matrix is

The blue strip represents the elements that are non zero.All other element in white area are of zero value.

I write down the code in java to represent non zero elements.

 import org.ujmp.core.Matrix;

public class Omegatest {
public static void main(String args[])throws Exception{
    Matrix omega= Matrix.Factory.zeros(2000,2000);
    Matrix Bigomega=Matrix.Factory.zeros(2000,2000);
    for(int k1=0;k1<1999;k1++){
        omega.setAsDouble(1, k1, k1);
        omega.setAsDouble(-1, k1, k1 + 1);
        omega.setAsDouble(-1, k1 + 1, k1);
        omega.setAsDouble(1, k1 + 1, k1 + 1);
        Bigomega = Bigomega.plus(omega);
        omega.clear();
    }
    System.out.println(Bigomega.inv());

}

}

Now whenever I try to invert this matrix, it gives me an error that the matrix is singular.

If I slightly change the code and take two Sparse Matrix instead of normal matrix it gives me a big value like 1902.003 1903.005 in each cells.

I know there is a definite problem in the code.

But first I want to know the mathematical explanation of this two conditions.

Thank you.

Answer 1

The sum of the elements in each row (or column) is zero, hence the matrix is not invertible. The matrix looks like this $$ \pmatrix{ 1 & -1 \\ -1 & 2 & -1 \\ &\ddots& \ddots&\ddots\\ &&\ddots& \ddots&\ddots\\ &&&-1 & 2 & -1 \\ &&&&1 & -1 } $$

(It seems to be some sort of stiffness matrix for a differential equation with Neumann boundary conditions, where constant functions are in the null space.)