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Symmetric and Skew Symmetric Matrices

Symmetric and Skew Symmetric Matrices. Theorem and Proof. Symmetric and Skew – Symmetric Matrix. A square matrix A is called a symmetric matrix, if A T = A. A square matrix A is called a skew- symmetric matrix, if A T = - A.

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Symmetric and Skew Symmetric Matrices

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  1. Symmetric and Skew Symmetric Matrices Theorem and Proof

  2. 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.

  3. 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.

  4. 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.

  5. Theorem 2 Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

  6. 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.

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

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

  9. Solution Cont.

  10. 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.

  11. ASSESSMENT (Symmetric and Skew Symmetric Matrices)

  12. 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)

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

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