Orthogonal & Orthonormal Matrices

Linear Algebra for Machine Learning · Axiom Academy

Verifying Orthogonality & Key Properties A square matrix Q is orthogonal if its columns are orthonormal (perpendicular unit vectors). This means QᵀQ = I. Q is orthogonal if and only if: Worked Example: Verify Orthogonality Multiply the transpose by the original: We get the identity matrix! Therefore Q is orthogonal. If Q has orthonormal columns q₁, q₂, ..., qₙ, then: The (i,j) entry of QᵀQ is qᵢ · qⱼ (dot product): If i = j: qᵢ · qᵢ = 1 (unit vector) If i ≠ j: qᵢ · qⱼ = 0 (perpendicular) Key Properties of Orthogonal Matrices Instead of computing A⁻¹ = (1/det(A))adj(A), we just use: Multiplying by Q doesn't change vector lengths: Angles between vectors are preserved under orthogonal transformation. Property 5: Rows are also Orthonormal Since Qᵀ is also orthogonal if Q is, the rows of Q are orthonormal too. Examples of Orthogonal Matrices Any rotation matrix is orthogonal: Reflection across the y-axis is orthogonal: Why orthogonal matrices matter: Fast computation: Q⁻¹ = Qᵀ is instant Numerical stability: No loss of precision QR decomposition for solving linear systems Singular Value Decomposition (SVD) Rotation and rigid motion in 3D graphics

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