Suppose I have a matrix C, dimension m x n, m < n, and that rank(C) = m.
I wish to find a marix V, dimension n-m x n, such that the square matrix T = [C ; V] is nonsingular.
What is a good criterion to use to select a V and how can one find that V that meets the crtierion, bearing in mind that T will be used for caculations like T\A*T?
For example, suppose
>> C = [1 2 3 4 5;5 6 7 8 9];
Then one possiblity is to define V by the null space of C
>> V1=null(C)'; T1 = [C;V1]; rank(T1)
Is this guaranteed to work for all C from a theoretical standpoint? If so, is this appoach too limiting insofar as there are solutions for V that don't live in the null space of C? Is there a possibiltiy of problem with null(C) if C is close to not being full rank?
For this simple example, another solution can be found by inspection:
>> V2 = [zeros(3,2) eye(3)];T2 = [C;V2]; rank(T2)
V2 has some appeal because it looks "simple" with only three non-zero elements that are all untiy. But can such a "simple" matrix be found when the C doesn't easily lend itself to inspection?
Neither solution looks all that appealing with respect to rcond
>> [rcond(T1) rcond(T2)]
though maybe these aren't all that bad.
In summary, are there any suggestions on how to find V?