Symmetric and Skew Symmetric Matrices Theorem and Proof

Symmetric and Skew – Symmetric Matrix A square matrix A is called a symmetric matrix, if AT = A. A square matrix A is called a skew- symmetric matrix, if AT = - A. Any square matrix can be expressed as the sum of a symmetric and a skew- symmetric matrix.

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′)′ (Why?) =A′ – A (Why?) = – (A – A ′) = – C Therefore C = A – A′ is a 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 writen as 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 is symmetric matrix and is skew symmetric matrix. Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

Example Show that A= is a skew-symmetric matrix. Solution : As AT = - A, A is a skew – symmetric matrix

Example Express the matrix as the sum of a symmetric and a skew- symmetric matrix. Solution :

Solution Cont.

Solution Cont. Therefore, P is symmetric and Q is skew- symmetric . Further, P+Q = A Hence, A can be expressed as the sum of a symmetric and a skew -symmetric matrix.

ASSESSMENT (Symmetric and Skew Symmetric Matrices)

Express the matrix as the sum of a Question 1: Express the matrix as the sum of a symmetric and askew symmetric matrix. Question 2: Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (II) (I) (III)

For the matrix , verify that (i) (A + A′) is a symmetric matrix Question 3: For the matrix , verify that (i) (A + A′) is a symmetric matrix (ii) (A – A ′) is a skew symmetric matrix Question 4: