Abstract: By using the method of matrix equation equivalent
transformation, combined the properties of $2$-norm and $F$-norm and
their relationship with eigenvalue, this paper dealt with the upper
bound for perturbation of diagonalized non-singular matrix
eigenspaces. Upper bound was obtained for matrix
eigenspace $\|{\rm sin}\Theta\|_{F}$ conditioned by $\eta_{2}=\|{\bm A}^{-\frac{1}{2}}{\bm E}{\bm A}^{-\frac{1}{2}}\|_{2}<1$.
The final theorem is the extension of theorem $4. 1$ in $[2]$.

Key words: Frobenius-norm, perturbation bounds, eigenspace, Frobenius-norm, perturbation bounds