Introduction:
In linear algebra, eigenvalues and eigenvectors are important concepts when studying matrices. Eigenvalues represent the scalar values that are associated with eigenvectors. Eigenvectors are special vectors that do not change direction when a linear transformation is applied to them.

Given information:
Matrix A has an eigenvector [111 111 111] corresponding to the eigenvalue 0. The corresponding column matrix is [0 1 -1].

Explanation:
To find the other orthogonal eigenvector for the same eigenvalue, we can use the concept of orthogonality and the properties of eigenvalues and eigenvectors.

Orthogonality:
Orthogonal vectors have a dot product of zero. If two vectors are orthogonal, they are perpendicular to each other.

Process:
To find the other orthogonal eigenvector, we will follow these steps:

1. Normalize the given eigenvector:
- Divide each element of the eigenvector [111 111 111] by its magnitude to normalize it.
- The magnitude of the eigenvector can be calculated using the formula: ||v|| = sqrt(v1^2 + v2^2 + v3^2).
- After normalization, we get the normalized eigenvector: [1/sqrt(3) 1/sqrt(3) 1/sqrt(3)].

2. Find a vector that is orthogonal to the normalized eigenvector:
- The dot product of two orthogonal vectors is zero.
- Let's assume the other orthogonal eigenvector as [x y z].
- Taking the dot product of the normalized eigenvector and the assumed orthogonal eigenvector should give us zero.
- The dot product can be calculated using the formula: dot_product = v1*u1 + v2*u2 + v3*u3, where v and u are the vectors.
- Expanding the dot product equation, we get: (1/sqrt(3))*x + (1/sqrt(3))*y + (1/sqrt(3))*z = 0.

3. Solve the equation to find the values of x, y, and z:
- The equation (1/sqrt(3))*x + (1/sqrt(3))*y + (1/sqrt(3))*z = 0 can be rearranged as: x + y + z = 0.
- This equation implies that the sum of the coordinates of the orthogonal eigenvector should be zero.
- There can be infinitely many solutions for x, y, and z that satisfy this equation.
- One possible solution is x = 1, y = -1, z = 0.
- Therefore, the other orthogonal eigenvector for the eigenvalue 0 is [1 -1 0].

Conclusion:
The other orthogonal eigenvector for the eigenvalue 0, given the eigenvector [111 111 111], is [1 -1 0]. This solution is obtained by normalizing the given eigenvector and finding a vector that is perpendicular to it. Orthogonal eigenvectors are important in various applications of linear algebra, such as solving systems of linear equations and diagonalizing matrices.