📘 Chapter 4: Symmetric, Skew-Symmetric & Orthogonal Matrices

🔁 Symmetric Matrix

A matrix \( A \) is symmetric if: \[ A^T = A \]

This means the matrix is equal to its own transpose.

↔️ Skew-Symmetric Matrix

A matrix \( A \) is skew-symmetric if: \[ A^T = -A \]

Also, diagonal elements of a skew-symmetric matrix are always zero.

📐 Orthogonal Matrix

A square matrix \( Q \) is orthogonal if: \[ Q^T Q = Q Q^T = I \]

Orthogonal matrices preserve length and angles—essential in 3D geometry and computer graphics.

📊 Examples

Symmetric: \[ A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, \quad A^T = A \]

Skew-Symmetric: \[ B = \begin{bmatrix} 0 & -4 \\ 4 & 0 \end{bmatrix}, \quad B^T = -B \]

Orthogonal: \[ Q = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \quad Q^T Q = I \]

📝 Practice

Q: Let \[ M = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 3 & 0 \\ 2 & 0 & 1 \end{bmatrix} \] Is \( M \) symmetric?

Hint: Check whether \( M^T = M \)