Leave $ R ^ 3 $ denotes the Euclidean space,$ |. | $ denotes the usual norm. For a matrix $$ T = begin {matrix}

a_1 and a_2 and a_3 \

a_4 & a_5 & a_6 \

a_7 & a_8 & a_9 \

end {matrix} $$ How to find the desired $ M $ such that for each $ alpha in R ^ 3 $ $$ | T (α) | le M | alpha | $$

(a discovery $ e_1, e_2, e_3 $ which form a basis for $ R ^ 3 $ then we can have $ | T (e_i) | le A_i | e_i | $ , leave $ M_1 = max {A_1, A_2, A_3 } $

(b) Find $ e_1, e_2, e_3 $ that form an orthogonal basis of $ R ^ 3 $ then we can have $ | T (e_i) | le A_i | e_i | $ , leave $ M_2 = max {A_1, A_2, A_3 } $

(c) Leave $ t_1, t_2, t_3 $ denote the proper values of $ T $ , and let $ M_3 = max {t_1, t_2, t_3 } $

We can prove that $ M_1 $ , $ M_2 $ or $ M_3 $ the desired $ M $ We are looking for ?

Then in general conditions. Leave $ H_1, H_2 $ denotes two normed spaces, $ T $ is an operator of $ H_1 $ to $ H_2 $ With a slight modification, we can prove that $ M_1 $ , $ M_2 $ or $ M_3 $ the desired $ M $ We are looking for ?