Oxford Linear Algebra: Eigenvalues and Eigenvectors Explained

University of Oxford mathematician Dr Tom Crawford explains how to calculate the eigenvalues and eigenvectors of a matrix, with 2 fully worked examples.

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Test your understanding with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: https://www.proprep.uk/Academic/Downl...

You can also find several video lectures from ProPrep explaining the topic further here: https://www.proprep.uk/general-module...

And fully worked video solutions from ProPrep instructors are here: https://www.proprep.uk/general-module...

Watch other videos from the Oxford Linear Algebra series at the links below.

Solving Systems of Linear Equations using Elementary Row Operations (ERO’s):    • Oxford Linear Algebra: Elementary Row...  

Calculating the inverse of 2x2, 3x3 and 4x4 matrices:    • Oxford Linear Algebra: How to find a ...  

What is the Determinant Function:    • Oxford Linear Algebra: What is the De...  

The Easiest Method to Calculate Determinants:    • Oxford Linear Algebra: The Easiest Me...  

The video begins by introducing the eigenvalue equation which we are trying to solve, with a discussion of possible methods of solution. We see that the only way a non-zero eigenvector can be found is if the determinant of the characteristic matrix is zero, which gives us the characteristic equation, or characteristic polynomial. Solving this equal to zero gives the eigenvalues, which are then substituted back into the eigenvalue equation to give the corresponding eigenvectors.

The method is demonstrated first with a 2x2 matrix example, and then for a 3x3 matrix. In both cases we consider a general eigenvector before choosing one parameter to make the final vector as simple as possible.

Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-c...

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