There are two possibilities for a matrix:
If m = n, the matrix is square
If m ≠ n, the matrix is rectangular
Any square matrix that satisfies the condition
If A is a matrix, then a symmetric matrix will follow the
Aᵀ = A,
where AT is the transpose of the matrix A. Transpose of a matrix means that if interchange the row and column, the new matrix obtained as result will be called the transpose of the original matrix. It also implies the condition,
A⁻ᵀ Aᵀ = I
Where I is the identity matrix.
A symmetric part of any matrix can be obtained by the following formula:
A₅ = ½ (A + Aᵀ)
Only a square matrix can be a skew-symmetric matrix. A skew-symmetric matrix is the one whose transpose is equal to the negative of itself.
Aᵀ = - A
A skew-symmetric part of any matrix can be obtained by the following formula:
Aₛₛ = ½ (A - Aᵀ)
Theorem 1: For any square matrix A with real number elements, A + Aᵀ is a symmetric matrix, and A - Aᵀ is a skew-symmetric matrix.
Let C = A + Aᵀ.
Taking a Transpose, Aᵀ = (A + Aᵀ)ᵀ = Aᵀ + (Aᵀ)ᵀ = A + Aᵀ = A + Aᵀ = C
This implies A + Aᵀ is a symmetric matrix.
Next, let B = A - Aᵀ
Bᵀ = (A + (-Aᵀ))ᵀ = Aᵀ + (-Aᵀ)ᵀ = Aᵀ - (Aᵀ)ᵀ = Aᵀ- A = - ( A - Aᵀ ) = - B
This implies A - Aᵀ is a skew-symmetric matrix.
Theorem 2: Any square matrix can be expressed as the sum of a skew symmetric matrix and a symmetric matrix. To find the sum of a symmetric and skew symmetric matrix, we use this formula:
Let B be a square matrix. Then,
B = (1/2) × (B + Bᵀ) + (1/2) × (B - Bᵀ). Here, Bᵀ is the transpose of the square matrix B.
If B + Bᵀ is a symmetric matrix, then (1/2) × (B + Bᵀ) is also a symmetric matrix
If B - Bᵀ is a skew symmetric matrix, then (1/2) × (B - Bᵀ) is also a skew symmetric matrix
Thus, any square matrix is eligible to be expressed as the sum of a skew symmetric matrix and a symmetric matrix.