1 / 13

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.

chun
Download Presentation

Symmetric and Skew Symmetric Matrices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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:

More Related