Eigenvalues and Eigenvectors in Linear Algebra
Learn about eigenvalues and eigenvectors in linear algebra, including the characteristic equation of a square matrix, finding eigenvalues, and corresponding eigenvectors. Explore examples and see how to solve systems of linear equations using eigenvectors.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Definition2 The equation det called the characteristic equation of a square matrix A. Theorem The eigenvalues of a square matrix Aare the roots of the characteristic equation det Example Find the Eigenvalue of and Eigenvector of the matrix Solution the characteristic equation is det
The eigenvalues are. or , or Next, we will substitute each of the two eigenvalues into the matrix equation: For the system of linear equations is:
Notice that the matrix equation represents a degenerated system of two linear equations. Both equations are constant multiples of the equation Then we get the relation Let
Similarly , the eigenvectors corresponding to are all nonzero multiples of Note If A is any 2 2 matrix, then its characteristic equation is So that
