# Relationship between eigenvalues of $A-B$ and eigenvalues of $A^2-B^2$

Let us suppose that $A_{n}$ and $B_n$ are sequences of positive definite matrices satisfying

$$c \leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C$$

and

$$c \leq \lambda_{\min}(B_n)\leq \lambda_{\max}(B_n)\leq C$$

where $\lambda_{\min}$ and $\lambda_{\max}$ are minimal and maximal eigenvalues. Then, what's the relationship between $\|A_n-B_n\|_2$ and $\|A_n^2-B_n^2\|_2$? Is it true that

$$\|A_n-B_n\|_2 \leq \text{constant} \cdot \|A_n^2-B_n^2\|_2$$

where the $2$-norm of a matrix $M$ is the maximum of the absolute value of its minimal eigenvalue and the absolute value of its maximal eigenvalue.

The norm you consider is usually called operator norm, or the norm subordinated to the $\ell^2$ norm over ${\mathbb C}^n$ (or ${\mathbb R}_n$). The correct inequality for positive Hermitian matrices $A$ an $B$ is $$\|A-B\|_2\le\sqrt{\|A^2-B^2\|_2}.$$ See Exercise 110 of my additional list accompagnying my book Matrices, GTM 216 (Springer-Verlag).
• "The norm you consider is usually called ..." or the spectral norm, or the norm induced by the $\ell^2$ norm, or... – Mark Meckes Dec 4 '12 at 14:57