Orthonormal Columns vs Orthogonal Matrix

A matrix with orthonormal columns is not necessarily an orthogonal matrix — unless it is also square. Matrix with orthonormal columns: Means 𝐴 𝑇 𝐴 = 𝐼 A T A=I (the columns are unit length and perpendicular to each other). Orthogonal matrix: Must be square and satisfy 𝐴 𝑇 𝐴 = 𝐴 𝐴 𝑇 = 𝐼 A T A=AA T =I. If the matrix isn't square (more rows than columns), it can't be orthogonal, even if the columns are orthonormal. Summary: Square + orthonormal columns → orthogonal matrix ✅ Rectangular + orthonormal columns → not orthogonal matrix ❌ Would you like a quick example to make it super clear? 🎯 Alright, let's do a quick example! 🔥 Example 1: Rectangular matrix with orthonormal columns (NOT orthogonal) Columns are orthonormal: ✅ (they are unit vectors and perpendicular) But 𝐴 A is not square → NOT an orthogonal matrix ❌ In fact, 𝐴 𝑇 𝐴 = 𝐼 A T A=I, but 𝐴 𝐴 𝑇 AA T is not the identity matrix. Example 2: Square matrix with orthonormal columns (Orthogonal matrix) Columns are

saved by

• Ravi Tej Guntuku