Finding a good inverse for an ill-conditioned matrix transformation

1. How do you find the inverse of an ill-conditioned matrix?
2. How do you know if a matrix is ill-conditioned?
3. What condition on rank A is necessary and sufficient for a to have a left inverse?
4. How do you determine if a matrix is ill-conditioned Python?

How do you find the inverse of an ill-conditioned matrix?

For badly conditioned matrices, you might opt in the SVD-route to calculate the inverse: A=UΣVH⟹A−1=VΣ−1UH. If your N×N matrix A is actually badly conditioned, you still should be able to perform Σ−1.

How do you know if a matrix is ill-conditioned?

A matrix is ill-conditioned if the condition number is too large (and singular if it is infinite).

What condition on rank A is necessary and sufficient for a to have a left inverse?

A matrix A_m×n has a left inverse A_left⁻¹ if and only if its rank equals its number of columns and the number of rows is more than the number of columns ρ(A) = n < m. In this case A⁺A = A_left⁻¹A = I.

How do you determine if a matrix is ill-conditioned Python?

In particular, the 2-norm condition number of a matrix is the largest singular value divided by the tiniest singular value; if the smallest singular value is zero, the matrix is singular, and if the smallest singular value is very tiny relative to the largest singular value, you have ill-conditioning.