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Subsection B.2.2 Spectral Analysis
This section provides a quick-reference guide to translating mathematical definitions into scientific and engineering terms.
Table B.2.6. Taxonomy of Norms: Mathematics vs. Physical Sciences
|
Euclidean norm (equivalent to \(L_2\)-norm) |
Vectors |
\(\| \vec{x} \|_2\) |
\(\sqrt{\sum_i^n x_i^2}\) |
Distance metrics, linear regression |
| Frobenius norm |
Matrices |
\(\| A \|_F\) |
\(\sqrt{\sum_{i=1}^m \sum_{j=1}^{n}A^2_{ij}}\) |
Matrix magnitude optimization, loss functions, and PCA |
| Physics Norm |
Vectors and Matrices |
🤔 |
\(\sqrt{A^T W A}\) where \(W\) is a weight matrix
|
Finite element analysis, fluid dynamics, and control systems |
Table B.2.7. Spectral Density and "Energy" Terms
| Energy |
Total squared variance over time |
1 J \(=\) 1 kg·m2⁄s2
|
| Power |
Time-averaged variance |
1 W \(=\) J⁄s
|
| Density |
Quantity distributed over a continuous domain |
Quantity per unit of that domain (e.g., W⁄Hz ) |
