Symmetric and Skew Symmetric Matrices

A square matrix A = [aij] is said to be symmetric if the transpose of A is equal to A, that is, [aij] = [aji] for all possible values of i and j.

A square matrix A = [aij] is a skew-symmetric matrix if A′ = – A, that is aji = – aij for all possible values of i and j. Also, if we substitute i = j, we have aii = – aii and thus, 2aii = 0 or aii = 0 for all i’s. Therefore, all the diagonal elements of a skew symmetric matrix are zero.

Properties of Symmetric Matrix

  • Addition and difference of two symmetric matrices results in symmetric matrix.
  • If A and B are two symmetric matrices and they follow the commutative property,

i.e. AB =BA, then the product of A and B is symmetric.

  • If matrix A is symmetric then An is also symmetric, where n is an integer.
  • If A is a symmetrix matrix then A-1 is also symmetric.

Properties of Skew Symmetric Matrix

  • When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.
  • Scalar product of skew-symmetric matrix is also a skew-symmetric matrix.
  • The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero. 
  • When identity matrix is added to skew symmetric matrix then the resultant matrix is invertible.
  • The determinant of skew symmetric matrix is non-negative

Theorem :  How do you prove symmetric and skew symmetric matrix?

Example

Properties of Symmetric and Skew-Symmetric Matrices

There are some rules that come from the concept of Symmetric and Skew-Symmetric Matrices,

1. If matrix A is a square matrix then (A + At) is always symmetric.

Prove: 

To find if a matrix symmetric or not, first, we have to find the transposed form of the given matrix 

So, let’s find the transpose of (A + At)

= (A + At)t

= A+ (At)t

= A+ A   [here, (At)= A]

= (A + At)

So, this is the same as the given matrix, so it is symmetric.

2. If matrix A is a square matrix then (A – At) is always skew-symmetric.

Prove:

To find if a matrix skew-symmetric or not, first, we have to find the transposed form of the given matrix 

So, let’s find the transpose of (A – At)

= (A − At)t

= A− (At)t

= A− A  [here, (At)t = A]

= − (A − At

So, this form is the negative of the given matrix, so it is skew-symmetric.

Theorem 1:

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 write A = 1/2 (A + A′) + 1/2 (A − A′). From the Theorem 1, we know that (A + A′) is a symmetric matrix and (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 = [aij]mxm

We know that A + A' : Symmetric matrix

                      A - A' : Skew Symmetric matrix

½(A + A' ) + ½ (A - A')

= (½A + ½A') + (½A - ½A') Using kA = k(A)

½A + ½A = A.

Theorem 3: Inverse of matrix A is unique.

Proof:

Let B and C be two inverses of A.

i.e., AxB=BxA=I

and  AxC =CxA=1

Now, B=BI = B (AxC)  = ( BxA) X C      = I xC    =     C

So, inverses are unique.