how is it possible that A = A/2 + A/2 + A'/2 - A'/2 - Maths - Matrices

Question

how is it possible that A = A/2 + A/2 + A'/2 - A'/2??????

Santosh Kumar · Accepted Answer

Let A be a square matrix $A = \frac{A}{2} + \frac{A}{2} + \frac{A'}{2} - \frac{A'}{2} = (\frac{A}{2} + \frac{A'}{2}) + (\frac{A}{2} - \frac{A'}{2}) = \frac{1}{2} (A + A') + \frac{1}{2} (A - A') Let P = \frac{1}{2} (A + A') Then, P' = [\frac{1}{2} (A + A')]^{'} = \frac{1}{2} [A' + (A')'] = \frac{1}{2} (A' + A) = P Let Q = \frac{1}{2} (A - A') Then, Q' = [\frac{1}{2} (A - A')]^{'} = \frac{1}{2} [A' - (A')'] = \frac{1}{2} (A' - A) = - Q$ So, P is a symmetric matrix and Q is a skew â€“ symmetric matrix $∴ A = P + Q = \frac{1}{2} (A + A') + \frac{1}{2} (A - A') = (\frac{A}{2} + \frac{A}{2}) + (\frac{A'}{2} - \frac{A'}{2}) = A$ Every square matrix can be expressed as the sum of a symmetric and a skew - symmetric matrix