The jordan product of two n x n matrices A B is A xJ B12AB

The jordan product of two n x n matrices A & B is

A x^J B=1/2(AB +BA)

Prove the following properties are true or give a counterexample to show that the property does not always hold. All matrices below are assumed to be n x n, I is the identity matrix, and r is any scalar.

1. A x^J B = B x^J A

2. (A x^J B) x^J C= A x^J (B x^J C)

3.( A+B) x^J C= (A x^J C) + (B x^J C)

Solution

The jordan product of nXn matrices is defined as below

A X JB = 1/2(AB + BA)

1) Proving the first property

we need to prove A X JB = B X JA

A X JB = 1/2(AB + BA) [ using the identity above]

B X JA = 1/2(BA + AB) (from the same identity]

since RHS or Right hand side of both the expressions are same, we get

A X JB = B X JA

2) Proving the second property

(A X JB) X JC = A X J(BXJC)

Now taking the left hand side we get

[1/2(AB + BA)] * JC = 1/2*AB*JC + 1/2 * BA * JC

=> 1/4 * (ABC + CAB) + 1/4 * (BAC + CBA)

Now considering the right hand side we get

A X J(BXJC) = A X J(1/2*(BC + CB)) = 1/2*A*J(BC) + 1/2 * A * J(CB)

=> 1/4*(ABC + BCA) + 1/4 * (ACB + CBA)

Hence both left and right hand sides are NOT equal always, so it follows the second property will not always be true

3) Proving the third property

(A+B) x JC = (AxJC) + (BXJC)

LHS:

(A+B)xJC

=> 1/2*((A+B)C + C(A+B)

=> 1/2*(AC+BC + CA + CB)

RHS:

(A xJC) + (B xJC)

=> 1/2*(AC + CA) + 1/2*(BC + CB)

Hence LHS = RHS