Open Access

Geometric significance of eigenvalues and eigenvectors in linear algebra and their potential value in data analysis

Yingdi Li

Yingdi Li

International School, Huanghe Science and Technology CollegeZhengzhou, China
Search for this author on
Sciendo|Google Scholar
Li, Yingdi
  
Sep 24, 2025
Geometric significance of eigenvalues and eigenvectors in linear algebra and their potential value in data analysis's Cover Image
About this article
Previous Article
Next Article
Cite
Download Cover
Published Online: Sep 24, 2025
Received: Jan 17, 2025
Accepted: May 05, 2025
DOI: https://doi.org/10.2478/amns-2025-0985
Keywords
© 2025 Yingdi Li, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1.

The trajectory of linear transformation
The trajectory of linear transformation

Figure 2.

The geometric meaning of eigenvalues and eigenvectors
The geometric meaning of eigenvalues and eigenvectors

Figure 3.

The corresponding curve of equation(3)~(5)
The corresponding curve of equation(3)~(5)

Figure 4.

The trajectory of linear transformation
The trajectory of linear transformation

Figure 5.

The trajectory of linear transformation
The trajectory of linear transformation

Figure 6.

Eigenvalue of Principal component analysis
Eigenvalue of Principal component analysis

Figure 7.

Spectra of potassium lamps
Spectra of potassium lamps

Figure 8.

Spectrum of xenon lamps
Spectrum of xenon lamps

Figure 9.

Grid division results of Zachary network
Grid division results of Zachary network

The results of the first ten principal components

Principal component Eigenvalue Contribution rate/% Cumulative contribution/%
1 2847.83 51.43 51.43
2 2672.39 46.28 97.71
3 61.27 0.83 98.54
4 54.78 0.46 99.00
5 10.83 0.63 99.63
6 6.28 0.21 99.84
7 3.12 0.05 99.89
8 2.76 0.03 99.92
9 2.14 0.02 99.94
10 1.82 0.01 99.95

MSE of xenon lamp spectra before and after denoising

Mean square error Line 524 Line 596 Line 974
Before denoising 0.548 0.476 0.503
After denoising 0.134 0.108 0.114

Selected eigenvectors and node category

Node Eigenvector1 Eigenvector2 Node number Node Eigenvector1 Eigenvector2 Node number
1 -0.1038 0.0647 1 18 -0.1001 0.1498 1
2 -0.0405 0.0936 1 19 0.1632 -0.0583 2
3 0.0231 0.0418 2 20 -0.0128 0.0647 1
4 -0.0527 0.1039 1 21 0.1539 -0.0594 1
5 -0.2876 -0.1203 2 22 -0.1000 0.1497 2
6 -0.3178 -0.1986 1 23 0.1579 -0.0597 2
7 -0.3194 -0.2006 1 24 0.1546 -0.0601 2
8 -0.5219 0.1007 1 25 0.1528 -0.0641 2
9 0.0504 0.0138 2 26 0.1489 -0.0732 1
10 0.0915 0.0129 2 27 0.1863 -0.0892 2
11 -0.02769 -0.1208 1 28 0.1176 -0.0359 1
12 -0.2117 0.7549 2 29 0.0948 -0.0059 2
13 -0.1093 0.1647 2 30 0.1634 -0.0698 2
14 -0.0139 0.0654 2 31 0.0726 0.0139 2
15 0.1576 -0.0613 1 32 0.0976 -0.0281 1
16 0.1643 -0.0619 2 33 0.1194 -0.0381 2
17 -0.4218 -0.3576 1 34 0.1173 -0.0285 1

Eigenvalues of Laplacian matrix and C(i) of Zachary network

Eigenvalue C(i) Eigenvalue C(i) Eigenvalue C(i)
λ1 0.0043 ─ λ12 0.0257 0.0001 λ23 0.0582 0.0013
λ2 0.0107 0.0064 λ13 0.0257 0.0000 λ24 0.0637 0.0055
λ3 0.0138 0.0031 λ14 0.0314 0.0057 λ25 0.0708 0.0071
λ4 0.0157 0.0019 λ15 0.0348 0.0034 λ26 0.0812 0.0104
λ5 0.0198 0.0041 λ16 0.0376 0.0028 λ27 0.0843 0.0031
λ6 0.0216 0.0018 λ17 0.0403 0.0027 λ28 0.0916 0.0073
λ7 0.0228 0.0012 λ18 0.0429 0.0026 λ29 0.1164 0.0248
λ8 0.0241 0.0013 λ19 0.0431 0.0011 λ30 0.1368 0.0204
λ9 0.0256 0.0015 λ20 0.0439 0.0008 λ31 0.1672 0.0304
λ10 0.0256 0.0000 λ21 0.0532 0.0093 λ32 0.2179 0.0507
λ11 0.0256 0.0000 λ22 0.0569 0.0037 λ33 0.2308 0.0129
Language:
English
Publication timeframe:
1 times per year
Journal Subjects:
Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other
Journal RSS Feed