Question

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:

`[(4, -2),(3, -5)]`

Answer 1

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

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

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

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

A + A^T = `[(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)]`

P^T = `1/2[(8, 1),(1, -10)]^"T"`

= `1/2[(8, 1),(1, -10)]`

P^T = P

∴ P is symmetric matrix.

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

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

A – A^T = `[(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)]`

Q^T = `1/2[(0, -5),(5, 0)]^"T"`

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

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

Q^T = `- 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.