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)]`

Answer 1

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)]`

A + A^T = `[(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)]`

P^T = `1/2[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]^"T"`

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

= P

P^T = P

∴ P = `1/2("A" + "A"^"T")` is a symmetric maatrix.

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)]`

A – A^T = `[(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)]`

Q^T =  `1/2[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]^"T"`

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

Q^T = `1/2 xx -1[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`

Q^T = `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