If A and B are symmetric matrices of same order such that A B + B A = X and A B − B A = Y , then ( X Y ) T =

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Since X Y = ( A B + B A ) ( A B − B A ) = ( AB ) AB + ( BA ) ( AB ) − ( AB ) ( BA ) − ( BA ) ( BA ) Now ( X Y ) T = (( AB ) ⋅ ( AB ) ) T + ( BA ⋅ AB ) T − ( AB ⋅ BA ) T = ( AB ) T ⋅ ( AB ) T + ( AB ) T ⋅ ( BA ) T − ( BA ) T ( AB ) T − ( BA ) T ( BA ) T ​ = ( B ⊤ ⋅ A ⊤ ) ( B ⊤ ⋅ A ⊤ ) + ( B ⊤ ⋅ A ⊤ ) ⋅ ( A ⊤ B ⊤ ) − ( A ⊤ B ⊤ ) ( B ⊤ A ⊤ ) − ( A ⊤ B ⊤ ) ( A ⊤ B ⊤ ) Since, A & B are symmetric matrix. = ( BA ) ( BA ) + ( BA ) ( AB ) − ( AB ) ( BA ) − ( AB ) ( AB ) = ( BA − AB ) ( BA + AB ) = − YX

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