A function of the form where ad – bc ≠ 0 is sometimes called a fractional linear transformation or a bilinear transformation. I usually use the name Möbius transformation. In what sense are Möbius transformations linear transformations? They’re nonlinear functions unless b = c = 0. And yet they’re analogous …

Sept. 22, 2023, 10:55 a.m.

Given a square complex matrix A, the Jordan normal form of A is a matrix J such that and J has a particular form. The eigenvalues of A are along the diagonal of J, and the elements above the diagonal are 0s or 1s. There’s a particular pattern to the …

Let A be an n × n complex matrix. The numerical range of A is the image of x*Ax over the unit sphere. That is, the numerical range of A is the set W(A) in defined by W(A) = {x*Ax | x ∈ ℂn and ||x|| = 1} where x* …

The previous post discussed an algorithm developed by Charles Dodgson (better known as Lewis Carroll) for computing determinants. The key identity for proving that Dodgson’s algorithm is correct involves the Desnanot-Jacobi identity from 1841. The identity is intimidating in its symbolic form and yet easy to visualize. In algebraic form …

Charles Dodgson, better known by his pen name Lewis Carroll, discovered a method of calculating determinants now known variously as the method of contractants, Dodgson condensation, or simply condensation. The method was devised for ease of computation by hand, but it has features that make it a practical method for …

An elegant algebraic identity says If x is the vector [a b] and y is the vector [c d] then this identity can be written where the dot indicates the usual dot product. I posted this on Twitter the other day. Gram matrix Now suppose that x and y are …

Here’s something I found surprising: the powers of a 2×2 matrix have a fairly simple closed form. Also, the derivation is only one page [1]. Let A be a 2×2 matrix with eigenvalues α and β. (3Blue1Brown made a nice jingle for finding the eigenvalues of a 2×2 matrix.) If …

I recently noticed something in a book I’ve had for five years: the bibliography section ends with a histogram of publication dates for references. I’ve used the book over the last few years, but maybe I haven’t needed to look at the bibliography before. This is taken from Bernstein’s Matrix …

Let A be an n × n matrix over a field F. The cofactor of an element Aij is the matrix formed by removing the ith row and jth column, denoted A[i, j]. This terminology is less than ideal. The matrix just described is called the cofactor of Aij, but …

A few days ago I wrote that circulant matrices all have the same eigenvectors. This post will show that it follows that circulant matrices commute with each other. Recall that a circulant matrix is a square matrix in which the rows are cyclic permutations of each other. If we number …

A circulant matrix is a square matrix in which each row is a rotation of the previous row. This post will illustrate a connection between circulant matrices and the FFT (Fast Fourier Transform). Circulant matrices Color in the first row however you want. Then move the last element to the …

This post will look at rotating a matrix 90° and what that does to the determinant. This post was motivated by the previous post. There I quoted a paper that had a determinant with 1s in the right column. I debated rotating the matrix so that the 1s would be …

When you solve systems of linear equations, you probably use Gaussian elimination, even if you don’t call it that. You may learn Gaussian elimination before you see it formalized in terms of matrices. So if you’ve had a course in linear algebra, and you sign up for a course in …

One of the surprising things about linear algebra over a finite field is that a non-zero vector can be orthogonal to itself. When you take the inner product of a real vector with itself, you get a sum of squares of real numbers. If any element in the sum is …

Marcel Golay discovered two “perfect” error-correcting codes: one binary and one ternary. These two codes stick out in the classification of perfect codes [1]. The ternary code is a linear code over GF(3), the field with three elements. You can encode a list of 5 base-three digits by multiplying the …

There are five ways to parenthesize a product of four things: ((ab)c)d (ab)(cd) (a(b(cd)) (a(bc))d (a((bc)d) In a context where multiplication is not associative, the five products above are not necessarily the same. Maybe all five are different. This post will give two examples where the products above are all …

A surprising amount of linear algebra doesn’t depend on the field you’re working over. You can implicitly assume you’re working over the real numbers R and prove all the basic theorems—say all the theorems that come before getting into eigenvalues in a typical course—and all or nearly all of the …

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