Linear Algebra With Python Notebooks Chapter 12 Eigenvalues And

Leo Migdal

-Nov 17, 2025, 6:08 AM

linear algebra with python notebooks chapter 12 eigenvalues and

There was an error while loading. Please reload this page. An eigenvector of an n×nn \times nn×n matrix AAA is a nonzero vector xxx such that Ax=λxAx = \lambda xAx=λx for some scalar λ\lambdaλ. A scalar λ\lambdaλ is called an eigenvalue of AAA if there is a nontrivial solution xxx of Ax=λxAx = \lambda xAx=λx, such an xxx is called an eigenvector corresponding to λ\lambdaλ. Since the eigenvector should be a nonzero vector, which means: The column or rows of (A−λI)(A-\lambda I)(A−λI) are linearly dependent

(A−λI)(A-\lambda I)(A−λI) is not full rank, Rank(A)<nRank(A)<nRank(A)<n. (A−λI)(A-\lambda I)(A−λI) is not invertible. This section introduces eigenvalues and eigenvectors of a square matrix and explores some of their applications. The goal of this section is to dissect the action of linear maps into elements that are easy to visualize. Consider a linear map $T: \mathbb{R}^n\to \mathbb{R}^m$ defined by $\vec{x} \mapsto A\vec{x}$. Although $T$ moves $\vec{x}$ in a variety of directions, there are some vectors on which the action of $T$ is easy to understand.

For example, suppose $A = \begin{bmatrix} 3 & -2 \\ 1 & 0\\ \end{bmatrix}$, $\vec{u } = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$. Let’s compute $A\vec{u}$ We can see that the image of $\vec{u} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$ is We are interested in such special vectors that when transformed by matrix $A$, result in a scalar multiple of themselves. More generally, let $A$ be an $n\times n$ matrix. We seek non-trivial solutions to equations of the form:

If $\vec{u}\neq 0$ is a solution for some $\lambda$, we call $\lambda$ an eigenvalue and $\vec{u}$ an eigenvector of $A$ corresponding to $\lambda$. Let $A$ be an $n\times n$ matrix (i.e. a square matrix). A non-zero vector $\vec{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$ if Rewriting this equation, we see that $\vec{v}$ is a solution of the homogeneous system of equations where $I$ is the identity matrix of size $n$.

Non-trivial solutions exists only when the matrix $A-\lambda I$ is noninvertible (singular). That is, when $\operatorname{det}(A-\lambda I) =0$. Therefore, the eigenvalues are the roots of the characteristic polynomial Here are three examples that we will consider. In each case, we have pre-computed the eigenvalues and eigenvectors (check them yourself). Notice, for matrix $D$ there is one eigenvalue that has two associated eigenvectors.

This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods. The copyright of the book belongs to Elsevier. We also have this interactive book online for a better learning experience. The code is released under the MIT license. If you find this content useful, please consider supporting the work on Elsevier or Amazon! < 15.3 The QR Method | Contents | 15.5 Summary and Problems >

Though the methods we introduced so far look complicated, the actually calculation of the eigenvalues and eigenvectors in Python is fairly easy. The main built-in function in Python to solve the eigenvalue/eigenvector problem for a square array is the eig function in numpy.linalg. Let’s see how we can use it. TRY IT Calculate the eigenvalues and eigenvectors for matrix $A = \begin{bmatrix} 0 & 2\\ 2 & 3\\ \end{bmatrix}$. Download: This and various other Jupyter notebooks are available from my GitHub repo. License: Creative Commons Attribution-ShareAlike 4.0 International License (CA BY-SA 4.0)

This is a tutorial related to the L665 course on Machine Learning for NLP focusing on Deep Learning, Spring 2018 at Indiana University. The following material is based on Linear Algebra Review and Reference by Zico Kolter (updated by Chuong Do) from September 30, 2015. This means, many passages are literally copied, many are rewritten. I do not mark sections that are added or different. Consider this notebook a extended annotation of Kolter's (and Do's) notes. See also James E.

Gentle (2017) Matrix Algebra: Theory, Computations and Applications in Statistics. Second edition. Springer. Another good resource is Philip N. Klein (2013) Coding the Matrix: Linear Algebra through Applications to Computer Science, Newtonian Press. For an alternative tutorial on that topic see also the HMC Mathematics Online Tutorial on Eigenvalues and Eigenvestors.

There was an error while loading. Please reload this page. Let $A$ be a square matrix. A non-zero vector $\mathbf{v}$ is an eigenvector for $A$ with eigenvalue $\lambda$ if Rearranging the equation, we see that $\mathbf{v}$ is a solution of the homogeneous system of equations $$ \left( A - \lambda I \right) \mathbf{v} = \mathbf{0} $$

where $I$ is the identity matrix of size $n$. Non-trivial solutions exist only if the matrix $A - \lambda I$ is singular which means $\mathrm{det}(A - \lambda I) = 0$. Therefore eigenvalues of $A$ are roots of the characteristic polynomial $$ p(\lambda) = \mathrm{det}(A - \lambda I) $$ This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods. The copyright of the book belongs to Elsevier.

We also have this interactive book online for a better learning experience. The code is released under the MIT license. If you find this content useful, please consider supporting the work on Elsevier or Amazon! < CHAPTER 15. Eigenvalues and Eigenvectors | Contents | 15.2 The Power Method > We learned from last chapter that matrix $A$ apply to column vector $x$, that is $Ax$, is a linear transformation of $x$.

There is a special transform in the following form: Where $A$ is $n\times{n}$ matrix, $x$ is $n\times{1}$ column vector ($X\neq{0}$), and $\lambda$ is some scalar. Any $\lambda$ that satisfies the above equation is known as an eigenvalue of the matrix $A$, while the associated vector $x$ is called an eigenvector corresponding to $\lambda$.

Linear Algebra With Python Notebooks Chapter 12 Eigenvalues And

People Also Search

There Was An Error While Loading. Please Reload This Page.

(A−λI)(A-\lambda I)(A−λI) Is Not Full Rank, Rank(A)<nRank(A)<nRank(A)<n. (A−λI)(A-\lambda I)(A−λI) Is

For Example, Suppose \(A = \begin{bmatrix} 3 & -2 \\

If \(\vec{u}\neq 0\) Is A Solution For Some \(\lambda\), We

Non-trivial Solutions Exists Only When The Matrix \(A-\lambda I\) Is