Geometric Preservation
Chapter 3: Subspaces and Orthogonality — Length, Angle,& Inner-Product Preservation
From the book
Chapter 3: Subspaces and Orthogonality. In the chapter mind map this icon labels Length, Angle,& Inner-Product Preservation. The discussion below is excerpted and lightly edited from § Theorem: Inner-Product Preservation in Mathematics for AI and Machine Learning.
If $A\in\mathbb{R}^{N\times N}$ is real-valued orthonormal, then for any $\mathbf x,\mathbf y\in\mathbb{R}^{N\times 1}$,
What this drawing shows
What you see. Represents transformations that preserve selected structure such as length, angle, area, or inner products.
In the mind map. Chapter 3 — Length, Angle,& Inner-Product Preservation. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 3: Subspaces and Orthogonality — see the mind-map node Length, Angle,& Inner-Product Preservation.