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Question
Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:
`[(4, -2),(3, -5)]`
Solution
Let A = `[(4, -2),(3, -5)]`
AT = `[(4, 3),(-2, -5)]`
A + AT = `[(4, -2),(3, -5)] + [(4, 3),(-2, -5)]`
= `[(4 + 4, -2 + 3),(3 - 2, -5 - 5)]`
A + AT = `[(8, 1),(1, -10)]`
`1/2("A" + "A"^"T") = 1/2[(8, 1),(1, -10)]`
Let P = `1/2("A" + "A"^"T")`
= `1/2[(8, 1),(1, -10)]`
PT = `1/2[(8, 1),(1, -10)]^"T"`
= `1/2[(8, 1),(1, -10)]`
PT = P
∴ P is symmetric matrix.
A – AT = `[(4, -2),(3, -5)] - [(4, 3),(-2, -5)]`
= `[(4 - 4, -2 - 3),(3 + 2, -5 + 5)]`
A – AT = `[(0, -5),(5, 0)]`
`1/2("A" - "A"^"T") = 1/2[(0, -5),(5, 0)]`
Let Q = `1/2("A" - "A"^"T")`
= `1/2[(0, -5),(5, 0)]`
QT = `1/2[(0, -5),(5, 0)]^"T"`
= `1/2[(0, 5),(- 5, 0)]`
= `1/2 xx - 1[(0, 5),(- 5, 0)]`
QT = `- 1/2[(0, -5),(5, 0)]`
= – Q
∴ Q is a skew symmetric marix.
A =`1/2("A" + "A"^"T") + 1/2("A" - "A")^"T"`
Thus A is expressed as a sum of a symmetric and skew-symmetric matrix.
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