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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)]`.
Solution
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)]`
∴ AT = `[(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`
∴ A + AT = `[(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 – AT = `[(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 ...[∵ aij = aij]
and Q is a skew symmetric matrix. ...[∵ aij = – aij]
∴ 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)]`.
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