Eigenvectors and Eigenvalues of a Matrix | Linear Algebra

A thorough intro to eigenvectors, eigenvalues, and the characteristic equation and characteristic polynomial of a matrix. We define these terms and do many examples, including eigenvalues and eigenvectors of a 3x3 matrix in addition to several 2x2 examples. We'll go over how to find the eigenvalues of a matrix and how to use those to find the eigenvectors of a matrix. We'll finish with a complex eigenvalue example. Linear Algebra course: https://www.youtube.com/playlist?list=PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG Linear Algebra exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc Join Wrath of Math to get exclusive videos, music, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 0:00 Intro 0:24 Definition of Eigenvalue and Eigenvector 2:01 Eigenvectors are Nonzero 2:26 Geometric View of Eigenvalues 4:03 How to Verify x is an Eigenvector 5:38 Deriving the Characteristic Equation 8:27 Example of Characteristic Equation 9:10 Use Characteristic Equation to Find Eigenvalues 11:01 Characteristic Polynomial 12:20 Find Eigenvalues of a 3x3 Matrix 13:34 Eigenvectors of a 3x3 Matrix 16:24 Equivalent Eigenvalue Statements 17:25 Complex Eigenvalues 19:34 Conclusion ◉Textbooks I Like◉ Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Abstract Algebra: https://amzn.to/3IjoZaO Linear Algebra: https://amzn.to/43xAWEz Calculus: https://amzn.to/3PieD1M Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/3tsaEER Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: https://www.patreon.com/join/wrathofmathlessons ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon! Outro music is mine. You cannot find it anywhere, for now. Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu My Math Rap channel: https://www.youtube.com/channel/UCQ2UBhg5nwWCL2aPC7_IpDQ/featured