NLA MT25, Weyl's inequality
Flashcards
@State Weyl’s inequality.
Suppose:
- $A$ is a matrix
- $E$ is another matrix (a pertubation)
Then:
- For all $i$, $\sigma _ i(A + E) \in \sigma _ i (A) + [- \vert \vert E \vert \vert _ 2, \vert \vert E \vert \vert _ 2]]$
- As a special case, $ \vert \vert A \vert \vert _ 2 - \vert \vert E \vert \vert _ 2 \le \vert \vert A + E \vert \vert _ 2 \le \vert \vert A \vert \vert _ 2 + \vert \vert E \vert \vert _ 2$
- As a special case, For the eigenvalues of a symmetric matrix $A$, for all $i$, $\lambda _ i (A + E) \in \lambda _ i (A) + [- \vert \vert E \vert \vert _ 2, \vert \vert E \vert \vert _ 2]$
@Prove ∆weyls-inequality, i.e. that if
- $A$ is a matrix
- $E$ is another matrix (a pertubation)
then:
- For all $i$, $\sigma _ i(A + E) \in \sigma _ i (A) + [- \vert \vert E \vert \vert _ 2, \vert \vert E \vert \vert _ 2]]$
- As a special case, $ \vert \vert A \vert \vert _ 2 - \vert \vert E \vert \vert _ 2 \le \vert \vert A + E \vert \vert _ 2 \le \vert \vert A \vert \vert _ 2 + \vert \vert E \vert \vert _ 2$
- As a special case, For the eigenvalues of a symmetric matrix $A$, for all $i$, $\lambda _ i (A + E) \in \lambda _ i (A) + [- \vert \vert E \vert \vert _ 2, \vert \vert E \vert \vert _ 2]$
- As a special case, $ \vert \vert A \vert \vert _ 2 - \vert \vert E \vert \vert _ 2 \le \vert \vert A + E \vert \vert _ 2 \le \vert \vert A \vert \vert _ 2 + \vert \vert E \vert \vert _ 2$
- As a special case, For the eigenvalues of a symmetric matrix $A$, for all $i$, $\lambda _ i (A + E) \in \lambda _ i (A) + [- \vert \vert E \vert \vert _ 2, \vert \vert E \vert \vert _ 2]$
@todo