A second look at the matrix of the discrete Fourier transform. Proof that it is unitary. How to apply it to a function, and why it costs O(NlogN) vs O(N^2). Video. Notes.
Author Archives: joaquindrut
A very special unitary matrix
Review of some properties of unitary matrices. Orthogonality and completeness. A first look at the matrix of the discrete Fourier transform. Diagonalization and efficiency. Video. Notes.
The Dirac delta
A first look at the Dirac delta and an application with Green’s functions. Video. Notes.
Eigensystems and Inverses. Part 2.
Recap of Part 1. Orthogonality and completeness relations. What do they look like in the infinite-dimensional case of part 1? Video. Notes.
Eigensystems and Inverses
Spectral representation of the inverse of a matrix. An example in infinite dimensions and… a first appearance of a Green’s function! Video. Notes.
Eigenvectors and Eigenvalues. Part 2.
Remider of part 1. Matrix diagonalization. Spectral representation. Similarity transformations. Two applications. Video. Notes.
Eigenvectors and Eigenvalues. Part 1.
A first look at eigenvectors and eigenvalues. Definition, quick examples, and a theorem. More on this in part 2!Video. Notes.
Determinants. Part 2.
Two applications/connections of determinants: the inverse of a matrix and Cramer’s rule, and eigenvectors and eigenvalues. More on the latter in a future video!Video. Notes.
Determinants. Part 1.
How are determinants defined and calculated? What are their main properties? I talk about that here and add a little example at the end on Gaussian integrals in arbitrary dimensions. Video. Notes.
Gaussian integrals and Feynman’s trick
We go over how to calculate the basic Gaussian integral and then other integrals using Feynman’s parameter differentiation approach. There will be more videos coming up on multidimensional Gaussians!Video. Notes.