# How to establish a relation between the 2-norm condition number and the eigenvalues of a matrix $A$ (not-necessarily symmetric)?

A relation between the 2-norm condition number of a matrix $$A$$ and its eigenvalues is stated as $$\kappa_2(A) \geq {\max{|\lambda_i(A)|} \over \min {|\lambda_i(A)|} } \tag{1}$$ where $$A$$ is not necessarily symmetric.

If we take for instance $$A = \left[ \matrix{ 1 & 2 \cr 3 & 4 \cr} \right],$$ then $$\kappa_2(A) = 14.9330, \ \max |\lambda_i(A)| = 5.3723, \min |\lambda_i(A)| = 0.3723$$ and (1) can be verified as $${\max |\lambda_i(A)| \over \min |\lambda_i(A)| } = 14.4300$$

If $$A$$ is symmetric, then we can use its special properties and in fact, the result (1) holds as an equality.

$$\kappa_2(A) = {\sigma_\max(A) \over \sigma_\min(A)} = {|\lambda_\max(A)| \over |\lambda_\min(A)|}$$ (for symmetric matrices)

For the general case, how to establish the result (1)?

I attempted a proof using the property that $$\rho(A) \leq \Vert A \Vert_2$$ where $$\rho(A)$$ is the spectral radius of $$A$$.

This shows that $$\max|\lambda_i(A)| \leq \Vert A \Vert_2 \tag{2}$$ and next we need to show that $${1 \over \min|\lambda_i(A)|} \leq \Vert A^{-1} \Vert_2$$

We also note that $$\rho\left( A^{-1} \right) \leq \Vert A^{-1} \Vert_2$$

As noted in the comments below, the eigenvalues of $$A^{-1}$$ are the reciprocals of the eigenvalues of $$A$$.

Hence, it is immediate that $${1 \over \min{ | \lambda_i(A) |}} = \rho\left( A^{-1} \right) \leq \Vert A^{-1} \Vert_2 \tag{3}$$

Combining (2) and (3), the result (1) follows, viz.

$${\max |\lambda_i(A)| \over \min |\lambda_i(A)|} \leq \kappa_2(A)$$

In fact, this result holds true for the condition number of $$A$$ ($$\kappa(A)$$) with respect to any operator norm of $$A$$, since $$\rho(A) \leq \Vert A \Vert$$ for any operator norm.

The eigenvalues of $$A^{-1}$$ are the reciprocals of the eigenvalues of $$A$$. So $$\frac{1}{\min |\lambda_i(A)|}=\max |\lambda_i(A^{-1})|$$; thus your property finishes the proof.
• The result $\rho(A) \leq \Vert A \Vert$ holds for any operator norm of $A$. The same proof can be applied to establish a lower bound for the condition number of a matrix with respect to any operator norm in terms of the eigenvalues of $A$ (symmetric or non-symmetric). Thanks! Apr 14, 2022 at 12:53