Uploaded August 2022 | Updated September 2026, 2 weeks ago
University of Oxford mathematician Dr Tom Crawford explains how to calculate the eigenvalues and eigenvectors of a matrix, with 2 fully worked examples.
Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford
Test your understanding with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20Eigenvectors%20Eigenvalues%20and%20Diagonalization%20-%20workbook%20uk.pdf
You can also find several video lectures from ProPrep explaining the topic further here: proprep.uk/general-modules/all/linear-algebra/eigenvectors-eigenvalues-and-diagonalization/eigenvectors-and--eigenvalues-of-a-matrix/vid25751
And fully worked video solutions from ProPrep instructors are here: proprep.uk/general-modules/all/linear-algebra/eigenvectors-eigenvalues-and-diagonalization/eigenvectors-and--eigenvalues-of-a-matrix/vid25736
Watch other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w
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: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
University of Oxford mathematician Dr Tom Crawford explains how to calculate the eigenvalues and eigenvectors of a matrix, with 2 fully worked examples.
Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects: proprep.uk/info/TOM-Crawford
Test your understanding with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free here: proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20Eigenvectors%20Eigenvalues%20and%20Diagonalization%20-%20workbook%20uk.pdf
You can also find several video lectures from ProPrep explaining the topic further here: proprep.uk/general-modules/all/linear-algebra/eigenvectors-eigenvalues-and-diagonalization/eigenvectors-and--eigenvalues-of-a-matrix/vid25751
And fully worked video solutions from ProPrep instructors are here: proprep.uk/general-modules/all/linear-algebra/eigenvectors-eigenvalues-and-diagonalization/eigenvectors-and--eigenvalues-of-a-matrix/vid25736
Watch other videos from the Oxford Linear Algebra series at the links below.
Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): youtu.be/9pF__coVyEE
Calculating the inverse of 2x2, 3x3 and 4x4 matrices: youtu.be/VKOaG3Ogf9Q
What is the Determinant Function: youtube.com/watch?v=bLsBWVYSg0A
The Easiest Method to Calculate Determinants: youtu.be/qniUv4EZB0w
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: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here: beautifulequations.net/collections/tom-rocks-maths
