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 $[2]$.