Question

How many decompositions of a square matrix into a symmetric part and a skew-symmetric part are possible?

Options

• Infinitely many decompositions are possible.

• The decomposition is unique.

• No such decomposition is possible.

• Exactly two decompositions are possible.

Answer 1

Every square matrix has the decomposition \[A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)\]. The symmetric and skew-symmetric parts obtained in this way are unique.