# What are the implications when matrix's lowest eigenvalue is equal to 0?

I have a task to solve where only eigenvalues are given and I need to calculate a matrix condition number. The formula for it requires division by the lowest eigenvalue (which is zero). In such case the condition number cannot be calculated.

Can equation system be solved if the lowest matrix eigenvalue is 0?

It is worth mentioning that for a non-singular, square matrix, the condition number induced by the $$2$$-norm is $$\kappa_2(A)=\sigma_1/\sigma_n$$, where $$\sigma_1$$ and $$\sigma_n$$ denote the largest and smallest singular values of $$A$$, not the eigenvalues. These are not always the same as the eigenvalues of $$A$$ (although, if $$A$$ is singular, then $$A^TA$$ does not have full rank, and so we will have a singular value of $$0$$).