Question

Let `A` be an `n × n` matrix. Show `A` equals the sum of a symmetric and a skew symmetric matrix ?

(`M` is skew symmetric if `M = −M^T`. `M` is symmetric if `M^T = M`.)

Hint: Show that `1/2(A^T + A)` is symmetric and then consider using this as one of the matrices.

Answer 1

See explanation...

Explanation:

Given an `n xx n` matrix `A`, let:

`{ (B = 1/2(A + A^T)), (C = 1/2(A - A^T)) :}`

Then:

`B + C = 1/2(A + A^T) + 1/2(A-A^T) = A`

`b_(ij) = b_(ji) = 1/2(a_(ij)+a_(ji))`

`c_(ij) = 1/2(a_(ij)-a_(ji)) = -1/2(a_(ji)-a_(ij)) = -c_(ji)`

So `B` is symmetric and `C` is skew symmetric.