Uploaded November 2022 | Updated September 2026, 2 weeks ago
University of Oxford mathematician Dr Tom Crawford goes through a full proof of the Spectral Theorem. 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 of the content covered in the video 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-%20Inner%20Product%20Spaces%20-%20workbook%20uk.pdf
You can also find several video lectures from ProPrep explaining the Spectral Theorem here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/the-spectral-theorem/vid30947
And further videos explaining the Gram-Schmidt process are here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/gram-schmitt-process/vid27109
Finally, fully worked video solutions from ProPrep instructors are here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/the-spectral-theorem/vid30951
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
Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w
The video goes through a full proof of the Spectral Theorem, which states that every real, symmetric matrix, has real eigenvalues, and can be diagonalised using a basis of its eigenvectors.
The first part of the proof uses the eigenvalue equation to show that any eigenvalue is in fact equal to its complex conjugate, and thus is real.
The second part of the proof shows that a matrix similarity transformation using an orthogonal matrix exists, and results in a diagonal matrix. We first construct an orthonormal basis (where the first vector is an eigenvector) using the Gram-Schmidt process, and then use these vectors as the columns of our orthogonal matrix. Next, we show that the resulting similarity matrix is also symmetric. This then allows us to conclude that the first row and first column are diagonal as required. The final step is to use induction on the size of the matrix. Assuming the result is true for a (n-1) x (n-1) matrix, we use our earlier calculation to construct the final orthogonal matrix, and show that when it is used as a change of basis matrix the result is diagonal, as we wanted.
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 goes through a full proof of the Spectral Theorem. 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 of the content covered in the video 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-%20Inner%20Product%20Spaces%20-%20workbook%20uk.pdf
You can also find several video lectures from ProPrep explaining the Spectral Theorem here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/the-spectral-theorem/vid30947
And further videos explaining the Gram-Schmidt process are here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/gram-schmitt-process/vid27109
Finally, fully worked video solutions from ProPrep instructors are here: proprep.uk/general-modules/all/linear-algebra/inner-product-spaces/the-spectral-theorem/vid30951
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
Eigenvalues and Eigenvectors Explained: youtu.be/8uISh6xyW7w
The video goes through a full proof of the Spectral Theorem, which states that every real, symmetric matrix, has real eigenvalues, and can be diagonalised using a basis of its eigenvectors.
The first part of the proof uses the eigenvalue equation to show that any eigenvalue is in fact equal to its complex conjugate, and thus is real.
The second part of the proof shows that a matrix similarity transformation using an orthogonal matrix exists, and results in a diagonal matrix. We first construct an orthonormal basis (where the first vector is an eigenvector) using the Gram-Schmidt process, and then use these vectors as the columns of our orthogonal matrix. Next, we show that the resulting similarity matrix is also symmetric. This then allows us to conclude that the first row and first column are diagonal as required. The final step is to use induction on the size of the matrix. Assuming the result is true for a (n-1) x (n-1) matrix, we use our earlier calculation to construct the final orthogonal matrix, and show that when it is used as a change of basis matrix the result is diagonal, as we wanted.
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
