Question

Symmetric And Skew-Symmetric Matrices

Answer 1

Symmetric matrix :

A square matrix A = |n,,] is called a symmetric matrix, if a_ij = a_ji for all i,j.

Example :

a_ij = a_ji for all i, j

⇒ (A)_ij = (A')_ij for ail i, j

⇒ A = A''

Skew-Symmetric Matrix :

A sqaure matrix A = [a_ij] is a skew symmetric matrix if A' = - A, that is a_ij = -a_ji or a_ij values of i and j.

a₁₂ = 2, a₂₁ = -2 ⇒ a₁₂ = -a₂₁

a₁₃ = -3, a₃₁ = 3 ⇒ a₁₃ = -a₃₁

and a₂₃ = 5, a₃₂ = -5 ⇒ a₂₃ = -a₃₂

It follows from the definition of a skew-symmetric matrix that A is skew-symmetric iff

⇒ a_ij = - a_ij for all i, j

⇒ (A)_ij = “ (A')_ij for all i, j

⇒ A = - A'

Thus, a square matrix A is a skew-symmetric matric if A' = -A.

Now, we are going to prove some results of symmetric and skew-symmetric matrices.

Theorem 1.

For any square matrix A with real number entries, A + A' is a symmetric matrix and A - A' is a skew- symmetric matrix.

Proof:

Let B = A + A', then

B' = (A + A')'

- A' + (A')' [As (A + B)' = A' + B']

= A' + A [As (A')' = A]

= A + A' [As A +B = B + A]

= B

Therefore B = A + A' is a symmetric matrix.

Now let C = A - A'

C'= (A - A')' = A' - (A')'

= A' - A = -(A - A') = -C

Therefore C = A - A' is skew-symmetric matrix.

Theorem 2.

Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Proof:

Let A be a square matrix, then we can write

A = \(\frac{1}{2}\) (A + A') + \(\frac{1}{2}\) (A - A')

From the Theorem 1, we know that (A + A') is a symmetric matrix and (A - A') is a skew symmetric matrix.

Since, for any matrix A, (kA)' = kA', it follows that \(\frac{1}{2}\) (A + A') is a symmetric matrix and \(\frac{1}{2}\) (A - A') is a skew symmetric matrix. Thus, any square matrix can be expressed as the sum of the symmetric and a skew symmetric matrix.