A first look at how linear algebra shows up in elementary machine learning. If you are new to linear algebra, you may find this interesting! Check it out! Video. Notes.
Category Archives: Linear algebra
Toward QMC: Spacetime determinant identities
I show how to rewrite the fermionic and bosonic determinants of previous videos in a blown-up spacetime formulation, where the matrices now become sparse and the connections to QFT and entanglement become much more interesting and useful! Check it out. Video. Notes.
Toward QMC: Trace-determinant identities. Part 2.
This is the fourth episode of the “Toward QMC” series. In the previous episode, I started to show how a crucial identity is proven in the case of fermions. Here I give more details on how to carry out that proof for products of multiple operators and explain what happens in the bosonic case. Check …
Continue reading “Toward QMC: Trace-determinant identities. Part 2.”
Toward QMC: Trace-determinant identities. Part 1.
This is the third episode of the “Toward QMC” series. In the previous episode, I brought up a trace-determinant identity without proof. Here I take a big first step to show how that identity is proven in the case of fermions. More on this (including bosons!) in part 2! Video. Notes.
The quantum free particle
Using background on complex numbers and Fourier modes, here is a first discussion of the most essential quantum mechanical system. Video. Notes.
The adjoint operator
A quick look at the adjoint of a linear operator (aka the hermitian conjugate). An example with a differential operator and how to prove properties using the definition based on the inner product. Video. Notes.
A 1d Green’s function example two ways
A closer look at how to calculate a Green’s function in a 1D example, using a direct solution and a spectral representation. Video. Notes.
Diagonalizing differential operators on the lattice
How do you represent a function on a computer with discrete memory? What about its derivatives? A first look at stuff on a lattice, the matrix form of the first-derivative operator (forward, with periodic boundary conditions), and its eigenvectors and eigenvalues (Fourier modes). Video. Notes.
A very special unitary matrix. Part 2.
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.
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.