Olga Taussky-Todd and John Francis are two of the most recognizable figures in matrix analysis. Both were directly involved with the study of flutter (dynamic instability) of aircraft wings, while working in London. Taussky worked at the National Physical Laboratory, during World War II, and Francis worked at the National Defense Research Corporation, in the late 50s.

Studying flutter leads to delicate boundary problems in partial differential equations, but R.A. Frazer recognized its close connection with the problem of locating the eigenvalues of a matrix.

In abstract, a useful theoretical tool to attack this problem is Geršgorin's theorem, though its efficiency may vary wildly from matrix to matrix: Suppose $A$ is an $n\times n$ matrix with complex entries $a_{ij}$, $1\le i,j\le n$. If, for each $i$, we let $r_i=\sum_{j=1,j\ne i}^n|a_{ij}|$, then for each eigenvalue $\lambda$ of $A$ there is an $i$ such that $|\lambda-a_{ii}|\le r_i$. That is, the eigenvalues of $A$ are located in the union of the $n$ closed discs centered at the diagonal entries of $A$, $\{z\in\mathbb C\mid |z-a_{ii}|\le r_i\}$, $1\le i\le n$.

Taussky writes:

A large group of young girls, drafted into war work, did the calculation on hand-operated machines, following the instructions of Frazer and his assistants.

As described in section 6 of *How I became a torchbearer for matrix theory*, Geršgorin's theorem proved indeed to be a key tool in carrying out the relevant computations.

Once again, I didn't ask to be assigned to matrix problems. They found me.

By the time Francis came to work on the problem, the connection with computation of eigenvalues was well established, and Francis was assigned to write the relevant computer programs to carry out this task. Trying to accelerate the time the computations required, he created the *shifted $QR$-algorithm* which, even nowadays, is one of the most efficient tools to compute eigenvalues (and, via companion matrices, roots of arbitrary polynomials).

As David Watkins writes in *Francis's algorithm*:

[Francis's method] became and has continued to be the big workhorse of eigensystem computations. A version of Francis’s algorithm was used by MATLAB when I asked it to
compute the eigensystem of a $1000\times 1000$ matrix on my laptop.