12 - Ostrowski

Ostrowski

Let $A,B\in {\mathbb{R}}^{n\times n}$
Let $A=(a_{ij})$ and $B=(b_{ij})$ for $1\le i,j\le n$

Eigenvalues of $A$ are continuously dependent on $(a_{ij})$

Let

$$\{{\lambda }_i,i=1,\cdots ,n\}\text{ and }\{{\mu }_i:i=1,\cdots ,n\}$$

be the eigenvalues of $A$ and $A+B$ respectively

For any $ϵ>0$ there exists $\delta >0$ such that

$$\underset{i,j}{\max }|b_{ij}|<\delta ⟹|{\lambda }_i-{\mu }_i|<ϵ$$