Symmetric and Skew Symmetric Matrices

Theorem and Proof

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.

Symmetric and Skew – Symmetric Matrix

T TA + A A - AA = + ,

2 2

T Twhere A + A is symmetric matrix and A - A is skew - symmetric matrix.

For any square matrix A with real number entries,

A + A ′ is a symmetric matrix and

A – A ′ is a skew symmetric matrix.

Theorem 1

Let B = A + A ′, thenB′ =(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 matrixNow let C = A – A′C′ = (A – A′ )′ = A ′ – (A′)′ (Why?)=A′ – A (Why?)= – (A – A ′) = – CTherefore C = A – A′ is a skew symmetric matrix.

Proof

Theorem 2

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

0 5 3

-5 0 -8

-3 8 0

Solution :

T

0 -5 -3 0 5 3

A = 5 0 8 =- -5 0 -8 =-A

3 -8 0 -3 8 0

Example

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

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

6 1 -5

A= -2 -5 4

-3 3 -1

Solution :

Example

6 1 -5

A = -2 -5 4

-3 3 -1

T

6 -2 -3

A = 1 -5 3

-5 4 -1

T

6 1 -5 6 -2 -31 1

Let P = (A+A )= -2 -5 4 + 1 -5 32 2

-3 3 -1 -5 4 -1

12 -1 -81

P = -1 -10 72

-8 7 -2

Solution Cont.6 -1/2 -4

P = -1/2 5 7/2

-4 7/2 -1

T

6 1 -5 6 -2 -31 1

Let Q = A- A = -2 -5 4 - 1 -5 32 2

-3 3 -1 -5 4 -1

T6 -1/2 -4

P = -1/2 5 7/2 =P

-4 7/2 -1

0 3 -21

Q = -3 0 -12

2 -1 0

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.

Solution Cont.0 3/2 -1

Q = -3/2 0 1/2

1 -1/2 0

T0 3/2 -1

Q =- -3/2 0 1/2 =-Q

1 -1/2 0

T TP = P and Q = -Qa

(Symmetric and Skew Symmetric Matrices)

ASSESSMENT

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:

(I) (II) (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: