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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

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.


Show that A= is a skew-symmetric matrix.

Example

Solution :

As AT = - A, A is a skew – symmetric matrix


Express the matrix as the sum of a

symmetric and a skew- symmetric matrix.

Example

Solution :



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)


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)


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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