Symmetric matrix - Example, Skew symmetric matrix Properties and theorems

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)

Properties of symmetric matrix

• Adding or subtracting two symmetric matrices results in a symmetric matrix
• They follow the commutative property I.e., AB = BA
• The product of two symmetric matrix is also symmetric matrix
• If a matrix A is symmetric then An will also be a symmetric matrix, where n is an integer
• If a matrix A is symmetric then A-1 will also be a symmetric matrix

Skew-symmetric matrix

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)

Properties of skew-symmetric matrix

• Adding or subtracting two skew-symmetric matrices results in a skew-symmetric matrix
• They follow the commutative property I.e., AB = BA
• The scalar product of two skew-symmetric matrix is also skew-symmetric matrix
• If an identity matrix is added to a skew-symmetric matrix, then the resultant is invertible
• The determinant of a skew-symmetric matrix is positive

Symmetric matrix theorems

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.

Proof:

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.