Square Matrix
Chapter 2: Matrix — Matrix Inverse ($A^{-1}$) & Singular Values
From the book
Chapter 2: Matrix. In the chapter mind map this icon labels Matrix Inverse ($A^{-1}$) & Singular Values. The discussion below is excerpted and lightly edited from § Definition: Singular Values and Vectors in Mathematics for AI and Machine Learning.
For a matrix $A \in \mathbb{R}^{M \times N}$ (which can be rectangular, i.e., $M \ne N$, or square), the singular values of $A$, denoted $\sigma_0(A), \sigma_1(A), \ldots, \sigma_{\min(M,N)-1}(A)$, are the square roots of the eigenvalues of the Gram matrix $A^\top A$ (or equivalently, of $AA^\top$). (Gram matrices are formally defined in Chapter the referenced section.)
Specifically, if $\lambda_0 \ge \lambda_1 \ge \cdots \ge \lambda_{r-1} > 0$ are the positive eigenvalues of $A^\top A$, where $r = \mathrm{rank}(A)$, then the singular values are defined as with $\sigma_i(A) = 0$ for $i \ge r$. The singular values are ordered such that $\sigma_0(A) \ge \sigma_1(A) \ge \cdots \ge \sigma_{\min(M,N)-1}(A) \ge 0$.
The largest singular value $\sigma_0(A)$ is also called the spectral norm of $A$: $||A||_2 = \sigma_0(A)$ (the Schatten $\infty$-norm, Section the referenced section).
What this drawing shows
What you see. Matrix with the same number of rows and columns, representing endomorphisms and eigenvalue problems.
In the mind map. Chapter 2 — Matrix Inverse () & Singular Values. See From the book above for definitions, figures, and worked examples.
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Read the full definitions, figures, and worked examples in Chapter 2: Matrix — see the mind-map node Matrix Inverse () & Singular Values.
