Please help with matrix analysis?

Question

Question Details:

Let A be any square matrix

a) Show that A + A^t is symmetric and A - A^t is skew Symmetric

b) Prove that there is one and only one way to write A as the sum of a symmetric and skew symmetric matrix.

Answer 1

(a)  A matrix is symmetric if it equals its transpose.  Consider the transpose of the matrix A + AT :

(A + AT)T = AT + (AT)T = AT + A = A + AT

So the transpose of A + AT is equal to itself, A + AT.  Therefore A + AT is symmetric.

A matrix is skew symmetric if it equals the negative of its transpose.  Consider the transpose of the matrix A - AT:

(A - AT)T = AT - (AT)T = AT - A = - (A - AT)

So the transpose of A - AT is equal to the negative of itself, -(A-AT).  Therefore A - AT is skew symmetric.

(b) Write A as a sum of a symmetric matrix S and a skew symmetric matrix K:

A = S + K            (1)

Now write the transpose of equation 1:

AT = ST + KT = S - K            (2)

(using the definition of a symmetric and skew symmetric matrix)

Now add equation (1) and equation (2) then divide by 2 to find S:

S = (1/2) (A + AT)

Subtract equation (2) from equation (1) then divide by 2 to find K:

K = (1/2) (A - AT)

So we have proven that there is only one way to write A as a sum of a symmetric and a skew symmetric matrix - namely, the symmetric matrix must be (1/2) (A + AT) and the skew symmetric matrix must be (1/2) (A - AT).

________

And we can verify that these two matrices do indeed add to A:

(1/2) (A + AT) + (1/2) (A - AT) = 1/2 A + 1/2 AT + 1/2 A - 1/2 AT = A

and that they are symmetric and skew symmetric respectively (this follows from part a).