Symmetric & Skew-Symmetric Matrices

Topic 4

Symmetric Matrix

A matrix is called symmetric if its transpose is exactly equal to the original matrix. This means elements are mirrored across the main diagonal.

Condition: Aᵀ = A

2×2 Example:

1 2
2 3

→

1 2
2 3

Since A = Aᵀ, it is symmetric.

3×3 Example:

1 2 3
2 4 5
3 5 6

Check symmetry:
a₁₂ = a₂₁ (2 = 2)
a₁₃ = a₃₁ (3 = 3)
a₂₃ = a₃₂ (5 = 5)
So matrix is symmetric.

Skew-Symmetric Matrix

A matrix is called skew-symmetric if its transpose is equal to the negative of the matrix. This means elements across diagonal are negatives of each other.

Condition: Aᵀ = −A

2×2 Example:

0 2
-2 0

→

0 -2
2 0

Transpose gives negative of original matrix.

3×3 Example:

0 2 -1
-2 0 4
1 -4 0

Check condition:
a₁₂ = -a₂₁ (2 = -(-2))
a₁₃ = -a₃₁ (-1 = -1)
a₂₃ = -a₃₂ (4 = -(-4))
So matrix is skew-symmetric.

Important Results

Diagonal elements of skew-symmetric matrix are always 0
Only square matrices can be symmetric or skew-symmetric
Every square matrix can be written as sum of symmetric and skew-symmetric matrices
Symmetric matrices have equal elements across diagonal
Skew-symmetric matrices have opposite elements across diagonal