Question

矩阵证明对任意的n阶方阵A,存在一个对称矩阵B及一个反对称矩阵C,使得A=B+C,且这种分解是唯一的.即若A=B+C,A

Answer 1

B = (A+A')/2 ; B' = (A'+A)/2 = B

C = (A-A')/2 ; C' = (A'-A)/2 = -C

A = B+C

又设:A = B1+C1 ;其中:B1' = B1 ; C1' = -C1

A = B+C = B1+C1 ;

∴ C1-C = B-B1 = (B-B1)' = (C1-C)'= -C1+C

∴-C1+C = C1-C

从而:

C1 = C

B1 = B