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Question
Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
Solution
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)]`
A + AT = `[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`
`1/2("A" + "A"^"T") = 1/2[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`
Let P = `1/2("A" + "A"^"T")`
= `1/2[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`
PT = `1/2[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]^"T"`
= `1/2[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`
= P
PT = P
∴ P = `1/2("A" + "A"^"T")` is a symmetric maatrix.
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)]`
A – AT = `[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
`1/2("A" - "A"^"T") = 1/2[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
Let Q = `1/2("A" - "A"^"T")`
= `1/2[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
QT = `1/2[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]^"T"`
`1/2[(0, -5, -3),(5, 0, -6),(3, 6, 0)]`
QT = `1/2 xx -1[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
QT = `1/2 [(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
= – Q
∴ Q = `1/2("A" - "A"^"T")` is a skew symmetric matrix.
A = `1/2("A" + "A"^"T") + 1/2("A" - "A"^"T")`
Thus A is expressed as a sum of symmeric and a skew symmetric matrx
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