Question

Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`.

Answer 1

A square matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix as

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

Let A = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`

∴ A^T = `[(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`

∴ A + A^T = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] + [(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`

= `[(3 + 3, 3 - 2, -1-4),(-2 + 3, -2 - 2, 1 - 5),(-4 - 1, -5 + 1, 2 + 2)]`

= `[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`

Also, A – A^T = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] - [(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`

= `[(3 - 3, 3 + 2, -1+ 4),(-2 - 3, -2 + 2, 1 + 5),(-4 + 1, -5 - 1, 2 - 2)]`

= `[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`

Let P = `(1)/(2)("A" + "A"^"T")`

= `(1)/(2)[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`

and Q = `(1)/(2)("A" - "A"^"T")`

= `(1)/(2)[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`

∴ P is a symmetric matrix          ...[∵ a_ij = a_ij]

and Q is a skew symmetric matrix.  ...[∵ a_ij = –  a_ij]
∴ A = P + Q

∴ A = `(1)/(2)[(6, 1, -5),(1, -4, -4),(-5, -4, 4)] + (1)/(2)[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`.