Ahmed, Mansoor. 2021. “Eigen Decomposition”. Technologies In Industry 4.0. https://www.technologiesinindustry4.com/2021/12/eigen-decomposition.html.
Eigen decomposition is very important in linear algebra. It is a factorization of a matrix into a canonical form. The matrix is denoted in terms of its eigenvalues and eigenvectors. Merely diagonalizable matrices may be factorized in this manner. The decomposition is named spectral decomposition when the matrix being factorized is a normal symmetric matrix.
“An Intuitive Guide to Linear Algebra”. 2021. betterexplained.com. https://betterexplained.com/articles/linear-algebra-guide/.
Algebra, Linear, and MIT OpenCourseWare. 2021. “Linear Algebra”. MIT Opencourseware. https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm.
This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.
Bilal, Muhammad Abdullah. 2021. “Eigenvectors and Eigenvalues: A Deeper Understanding.” Medium. Analytics Vidhya. March 21. https://medium.com/analytics-vidhya/eigenvectors-and-eigenvalues-a-deeper-understanding-c715f8ded4c7.
The literal meaning of “Eigen” is characteristic which comes from the German language. These vectors are aptly named as these vectors ultimately give the trend something follows. Any matrix that has an eigenvector, all its coordinates will ultimately follow a relation as close to the eigenvector as possible. To expand on this I will be using some examples.
⭐ Bourne, Murray. 2024. “8. Applications of Eigenvalues and Eigenvectors.” Intmathcom RSS. Accessed June 13. https://www.intmath.com/matrices-determinants/8-applications-eigenvalues-eigenvectors.php.
Why are eigenvalues and eigenvectors important? Let’s look at some real life applications of the use of eigenvalues and eigenvectors in science, engineering and computer science.
⭐ “Eigenvectors And Eigenvalues”. 2019. napsterinblue.github.io. https://napsterinblue.github.io/notes/stats/lin_alg/eigenvectors_and_values/.
Following along with 3blue1brown’s series on The Essence of Linear Algebra, the topic of Eigenvectors and Eigenvalues shows up nearly last.
When I learned this in undergrad, it was a series of equations and operations that I memorized. However, revisiting to write this notebook, I’ve now got a good intuition for conceptualizing eigenvectors represent, as well as understand their use/role in Linear Algebra.
“Eigenvectors And Eigenvalues Explained Visually”. 2021. Explained Visually. https://setosa.io/ev/eigenvectors-and-eigenvalues/.
Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google’s PageRank algorithm. Let’s see if visualization can make these ideas more intuitive.
Erik Cheever, Swarthmore College. 2021. “Eigenvalues And Eigenvectors”. lpsa.swarthmore.edu. https://lpsa.swarthmore.edu/MtrxVibe/EigMat/MatrixEigen.html.
“Introduction To Eigenvalues And Eigenvectors (Video) | Khan Academy”. 2021. Khan Academy. https://www.khanacademy.org/math/linear-algebra/alternate-bases/eigen-everything/v/linear-algebra-introduction-to-eigenvalues-and-eigenvectors.
“Lecture 21: Eigenvalues And Eigenvectors | Video Lectures | Linear Algebra | Mathematics | MIT Opencourseware”. 2021. ocw.mit.edu. https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-21-eigenvalues-and-eigenvectors/.
If the product Ax points in the same direction as the vector x, we say that x is an eigenvector of A. Eigenvalues and eigenvectors describe what happens when a matrix is multiplied by a vector. In this session we learn how to find the eigenvalues and eigenvectors of a matrix.
“Eigen-WHAT?” 2021. Medium. https://medium.com/@dpedrazatrivino2/eigen-what-583125ddc984.
If you are reading this, you probably had a linear algebra class and remember your professor mentioning two strange words: Eigenvector and Eigenvalue. Maybe you remember the equation. However, WHAT ARE THEY?
Eigenvector and Eigenvalue. 2021. https://www.mathsisfun.com/algebra/eigenvalue.html.
“Eigenvalues And Eigenvectors”. 2021. Medium. https://towardsdatascience.com/eigenvalues-and-eigenvectors-378e851bf372.
I have learned about eigenvalues and eigenvectors in University in a linear algebra course. It was very dry and mathematical, so I did not get, what it is all about. But I want to present this topic to you in a more intuitive way and I will use many animations to illustrate it.
“Eigenvector And Eigenvalue”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/eigenvalue.html.
Free Linear Algebra textbook . 2021. https://joshua.smcvt.edu/linearalgebra/.
Malik, Farhad. 2020. “What Are Eigenvalues and Eigenvectors?” Medium. FinTechExplained. https://medium.com/fintechexplained/what-are-eigenvalues-and-eigenvectors-a-must-know-concept-for-machine-learning-80d0fd330e47.
If the new transformed vector is just a scaled form of the original vector then the original vector is known to be an eigenvector of the original matrix. Vectors that have this characteristic are special vectors and they are known as eigenvectors. Eigenvectors can be used to represent a large dimensional matrix.
Merwansky. “Understanding Eigenvectors and Eigenvalues”. 2023. Medium. https://just-merwan.medium.com/understanding-eigenvectors-and-eigenvalues-3bb839d0790c.
Eigenvectors and Eigenvalues are crucial concepts in linear algebra with practical applications in various fields. They allow us to understand and analyze complex systems, from vibrations and waves to machine learning algorithms, computer vision, image processing, and control systems.
⭐ Powell, Victor and Lewis Lehe. 2024. “Eigenvectors and Eigenvalues Explained Visually.” Explained Visually. Accessed June 13. https://setosa.io/ev/eigenvectors-and-eigenvalues/.
Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google’s PageRank algorithm. Let’s see if visualization can make these ideas more intuitive.
Soula, Walid. 2024. “Eigenvectors and Eigenvalues.” Medium. Artificial Intelligence in Plain English. June 6. https://ai.plainenglish.io/eigenvectors-and-eigenvalues-82c9ec34fd20.
An Eigenvalue (λ) of matrix A is a scalar value that satisfies a specific property: there exists a non-zero vector X such that when we multiply A by X, we get the same thing as multiplying X by λ. Mathematically, this can be represented as: “AX = λX” A vector X that satisfies the above equation is called an Eigenvector of matrix A corresponding to the Eigenvalue λ.
Note: The Eigenvector is not unique; it’s possible to have multiple Eigenvectors corresponding to the same Eigenvalue.
“Visualizing Eigenvalues And Eigenvectors”. 2019. Medium. https://towardsdatascience.com/visualizing-eigenvalues-and-eigenvectors-e2f9e3ac58d7.
Eigenvalues and Eigenvectors are a very important concept in Linear Algebra and Machine Learning in general. In my previous article, I’ve been introducing those concepts in terms of Principal Components Analysis, providing practical examples. In this article, I’m going to dwell more on the maths behind those concepts, providing a geometric interpretation of what I’m about to explain.
Ye, Andre. 2020. “The Jewel of the Matrix: A Deep Dive Into Eigenvalues & Eigenvectors.” Medium. Towards Data Science. June 22. https://towardsdatascience.com/the-jewel-of-the-matrix-a-deep-dive-into-eigenvalues-eigenvectors-22f1c8da11fd.
Eigenvalues and eigenvectors may seem abstract and distant concepts, but they play an integral role in the world around you. As everything becomes defined by data, they are stored in matrices. At the center of the matrix — through the complexities and complications — lie the eigenvectors and eigenvalues, providing clarity and revealing the nature of the matrix. Understanding what they are, how to derive them, and their applications are integral to appreciating the beauty of the matrix and, more broadly, the role data and mathematics play in the world.
Ming, Albert. “Part 1: Matrix Definitions”. 2022. Medium. https://albertming88.medium.com/part-1-matrix-definitions-7bb61c846d95.
Ming, Albert. “Part 2: Vectors”. 2022. Medium. https://albertming88.medium.com/part-2-vectors-bacbca5ad20.
Ming, Albert. “Part 3: Simple Matrices And Transformations”. 2022. Medium. https://albertming88.medium.com/part-3-simple-matrices-and-transformations-f90f43c99992.
Ming, Albert. “Part 4: Determinants”. 2022. Medium. https://albertming88.medium.com/part-4-determinants-b5a9ed88e40c.
Ming, Albert. “Part 5: Revisiting Systems Of Linear Equations”. 2022. Medium. https://albertming88.medium.com/part-5-revisiting-systems-of-linear-equations-ebe2beef6cd3.
Ming, Albert. “Part 6: Eigenvectors And Eigenvalues”. 2022. Medium. https://albertming88.medium.com/part-6-eigenvectors-and-eigenvalues-d94f507c4962.
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⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
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