Special Matrices and Symmetric Properties

Special Matrices and Symmetric Properties

This lesson covers the following key concepts:

• Symmetric matrix: A = Aᵀ
• Skew-symmetric matrix: A = -Aᵀ
• Expressing any matrix as sum of symmetric and skew-symmetric
• Orthogonal matrices: AAᵀ = I
• Properties of symmetric and skew-symmetric matrices
• Conjugate and complex matrices
• Hermitian and skew-Hermitian matrices

Important Formulas

• $$A = A^T \text{ (symmetric)}$$
• $$A = -A^T \text{ (skew-symmetric)}$$
• $$A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$$
• $$\text{Diagonal elements of skew-symmetric = 0}$$
• $$AA^T = I \Rightarrow |A| = \pm 1 \text{ (orthogonal)}$$
• $$A + A^T \text{ always symmetric}$$
• $$A - A^T \text{ always skew-symmetric}$$