[Zhan Xiang matrix theory exercise reference] exercise 1.13

13. (Li-Poon) proof: Each real matrix can be written as a linear combination of $4 $ real positive matrix, that is, if $ A $ is a real matrix, the real positive matrix $ q_ I $ and the real number $ r_ I $, $ I = 1, 2, 4 $, make $ \ Bex a = r_1q_1 + r_2q_2 + r_3q_3 + r_4q_4. \ EEx $

 

 

Proof: (1 ). first, it is proved that the spectral norm of $ A $ is the maximum singular value of $ A $. in fact, $ \ beex \ Bea \ Sen {A} _ \ infty ^ 2 & =\ Max _ {\ Sen {x} _ 2 = 1} \ Sen {ax} _ 2 ^ 2 \ & =\ Max _ {\ Sen {x} _ 2 = 1} x ^ * a ^ * ax \ & =\ Max _ {\ Sen {x }_ 2 = 1} x ^ * VV ^ * a ^ * u ^ * uavv ^ * x \ & =\ Max _ {\ Sen {y} _ 2 = 1} y ^ * \ diag (S_1 ^ 2, \ cdots, s_p ^ 2) Y \ quad \ sex {Y = V ^ * x} \ & =\ Max _ {\ Sen {y} _ 2 = 1} \ sum _ {I = 1} ^ P s_ I ^ 2 | y_ I | ^ 2 \ & = S_1 ^ 2. \ EEA \ eeex $(2 ). decomposed by singular values, there is a $ U, V $ makes $ \ Bex \ frac {1 }{\ Sen {A }_\ infty} UAV = \ diag (S_1, \ cdots, s_n), \ Quad 1 = S_1 \ geq S_2 \ geq \ cdots \ geq s_n \ geq 0. \ EEx $ if $ n = 2 K $, then the orthogonal array $ \ Bex R =\frac {1} {\ SQRT {2 }}\ sex {\ BA {CC}-1 & 1 \ 1 & 1 \ EA} \ EEx $ make $ \ Bex R ^ t \ sex {\ BA {CC} s _ {2j-1} & 0 \ 0 & S _ {2j} \ EA} R = \ sex {\ BA {CC} B _j & C_j \ C_j & B _j \ EA }, \ quad 2b_j = S _ {2j-1} + S _ {2j}, \ quad 2c_j = S _ {2j}-S _ {2j-1 }. \ EEx $ so, $ \ beex \ Bea & \ quad \ diag (R, \ cdots, R) ^ t \ diag (S_1, \ cdots, s_n) \ diag (R, \ cdots, R) \\\&=\ diag \ sex {\ BA {CC} B _1 & C_1 \ C_1 & B _1 \ EA }, \ cdots, \ sex {\ BA {CC} B _k & c_k \ c_k & B _k \ EA }\\\&=\ frac {1} {2} \ diag \ sex {\ BA {CC} B _1 & \ SQRT {1-b_1} \-\ SQRT {1-b_1} & B _1 \ EA }, \ cdots, \ sex {\ BA {CC} B _k & \ SQRT {1-b_k} \-\ SQRT {1-b_k} & B _k \ EA }\\ & \ quad + \ frac {1} {2} \ diag \ sex {\ BA {CC} B _1 &-\ SQRT {1-b_1 }\\ SQRT {1-b_1} & B _1 \ EA }, \ cdots, \ sex {\ BA {CC} B _k &-\ SQRT {1-b_k} \ SQRT {1-b_k} & B _k \ EA }\\ & \ quad + \ frac {1} {2} \ diag \ sex {\ BA {CC} \ SQRT {1-c_1} & C_1 \ C_1 &-\ SQRT {1-c_1} \ EA }, \ cdots, \ sex {\ BA {CC} \ SQRT {1-c_k} & c_k \ c_k &-\ SQRT {1-c_k} \ EA }\\ & \ quad + \ frac {1} {2} \ diag \ sex {\ BA {CC}-\ SQRT {1-c_1} & C_1 \ C_1 & \ SQRT {1-c_1} \ EA }, \ cdots, \ sex {\ BA {CC}-\ SQRT {1-c_k} & c_k \ c_k & \ SQRT {1-c_k} \ EA }}. \ EEA \ eeex $ then $ \ Bex \ frac {2} {\ Sen {A }_\ infty} \ diag (R, \ cdots, R) ^ t UAV \ diag (R, \ cdots, R) \ EEx $ is the sum of $4 $ real-forward arrays. if $ n = 2 k + 1 $, the certificate is issued, $ \ Bex \ diag (1, R, \ cdots, R) ^ t \ diag (1, S_2, \ cdots, s_n) \ diag (1, R, \ cdots, R) \ EEx $ is also half of the sum of $4 $ real-forward arrays. we have also concluded that.

[Zhan Xiang matrix theory exercise reference] exercise 1.13