Orthonormal Matrix
Chapter 3: Subspaces and Orthogonality — Column ($Q^\top Q = I$) vs. Square
From the book
Chapter 3: Subspaces and Orthogonality. In the chapter mind map this icon labels Column ($Q^\top Q = I$) vs. Square. The discussion below is excerpted and lightly edited from § Definition: Column Orthonormal Matrix in Mathematics for AI and Machine Learning.
Let $A = \begin{bmatrix}\mathbf a_0 & \mathbf a_1 & \cdots & \mathbf a_{N-1}\end{bmatrix} \in \mathbb{R}^{M\times N}$, $M \ge N$. Matrix $A$ is column orthonormal if the vectors $\{\mathbf a_0,\ldots,\mathbf a_{N-1}\}$ form an orthonormal basis of an $N$-dimensional subspace of $\mathbb{R}^M$.
What this drawing shows
What you see. Represents matrices whose columns or rows preserve lengths and angles.
In the mind map. Chapter 3 — Column () vs. Square. 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 Column () vs. Square.