Question

Show that every square matrix can be represented as the sum of a symmeric matrix and skew symmetric matrix

Answer 1

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Answer 2

Step-by-step explanation:

To prove uniqueness,let A=R+S where R is symmetric and S is skew−symmetric ∴ A'=(R+S)'=R'+S'=R−S { ∵ R'=R and S'=−S by Definition of symmetric and skew−symmetric matrices } ∴ 1/2(A+A')=1/2(R+S+R−S)=R=P

1/2(A−A')=1/2(R+S−R+S)=S=Q.

Hence, the representation A=P+Q is unique. Hence, it is proved that every square matrix can be uniquely expressed as a sum of symmetric and skew-symmetric matrix.