A Bird’s-Eye View of Linear Algebra: Orthonormal Matrices

This is the eighth chapter of the in-progress book on linear algebra: “A birds eye view of linear algebra”. The table of contents so far:

• Chapter-1: The basics
• Chapter-2: The measure of a map — determinants
• Chapter-3: Why is matrix multiplication the way it is?
• Chapter-4: Matrix chain multiplication
• Chapter-5: Systems of equations, linear regression and neural networks
• Chapter-6: Rank nullity and why row rank == col rank
• Chapter-7: Left-right inverse => injective-surjective maps
• Chapter-8 (current): Orthonormal matrices

In this chapter, we will cover special kinds of matrices: orthogonal and orthonormal. These kinds of matrices (and the corresponding linear maps they represent) have good properties, from theoretical to numerical that make them easy to work with. For instance, to get the inverse of an orthonormal matrix, you can simply flip it (take its transpose). But we’re getting ahead of ourselves. Let’s understand what it even means for a matrix to be orthonormal.