Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Lanczos240In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at].

Cornelius Lanczos was born in Székesfehérvár, Hungary in 1893. After his schooling there, he entered the University of Budapest in 1910, where he studied physics, mathematics and philosophy. His teachers included Lorand Eőtvős and Lipót Fejér. In his PhD studies, Lanczos aimed to combine general relativity and electromagnetic theory. An early link with Ireland was his use in his doctoral dissertation of quaternion calculus. Quaternions had been invented by Irish mathematician William Rowan Hamilton in 1843.


Charcoal sketch of Cornelius Lanczos in 1973 by John Chirnside of the Mechanical Engineering Department at UMIST.
By permission of the Royal Irish Academy © RIA.

Assisting Einstein

Following his doctoral studies, Lanczos moved to Freiburg, Germany and, in 1924, obtained a position as Assistant Professor of Physics in Frankfurt. There he produced an early formulation of quantum mechanics, closely related to Schrödinger’s Equation. Albert Einstein had read and approved of Lanczos’ doctoral dissertation. In 1928, Einstein offered Lanczos a one-year scholarship to work as his mathematical assistant in Berlin. In terms of joint publications, nothing emerged from this collaboration, but the two scientists remained on close terms despite differing scientific perspectives.

Lanczos moved to America in 1931, to a position at Purdue University in Indiana. Here he worked for the most part on a range of problems in applied mathematics. He devised an approximation technique called the tau method, which was used extensively in the Mathematical Tables Project, and for the preparation of the famous Handbook of Mathematical Functions, edited by Milton Abramowitz and Irene Stegun. This was for many years a standard sourcebook for applied mathematicians and engineers.

Returning briefly to Hungary in 1939, Lanczos managed to get his son to the United States on the Empress of Australia, the last ship to leave Europe for America before the outbreak of war. His wife, who was ill, was unable to travel, and she died shortly afterwards. In 1944, several members of his family were deported to Birkenau, where they perished along with countless other Jews.

Fast Fourier Transform

Lanczos was a brilliant lecturer and expositor of mathematics and physics. He wrote some excellent books, several of which are still in use. One of these was his “Discourse on Fourier Series”. Lanczos found an efficient way to compute Fourier transforms. This was twenty-five years before the publication by Cooley and Tukey of what is now called the Fast Fourier Transform (FFT). The FFT has had an enormous impact on computational methods, making many difficult problems practicable.


Opening lines from Lanczos’s monograph on Fourier Theory.

In 1944, Lanczos moved to the Boeing Aircraft Company, where he worked on practical design problems, and he devised several brilliant and powerful methods in numerical analysis. One of these – now called the Lanczos method – was for finding eigenvalues of large symmetric matrices. It has been very influential and is still in widespread use, for example in numerical weather prediction. But like his discovery of the FFT, it was forgotten for decades before attracting attention. It remains of central interest: today, the Lanczos method is one of the most popular methods for matrix computations. It was listed among the top ten numerical algorithms of the twentieth century.

To Dublin

In May 1954, Lanczos moved to Dublin as senior professor at DIAS. De Valera had been inspired by the Institute for Advanced Study in Princeton, established in 1930, to set up Dias, with Erwin Schrödinger as the first director of the School of Theoretical Physics.

In Dublin, Lanczos worked on both mathematics and physics. One of his early breakthroughs in mathematics was what is now known as the singular value decomposition (SVD). Lanczos showed that an arbitrary (rectangular) matrix can be factored into three parts, the first and last being orthogonal matrices and the middle component being a diagonal matrix. Lanczos published this work in 1958 and was awarded the Chauvenet Prize by the Mathematical Association of America. In fact, Lanczos was not the first to discover this factorization, but he contributed greatly to the popularization of the method. The result is today referred to as “The Fundamental Theorem of Linear Algebra”.

Unified Field Theory

While Lanczos tackled applied mathematics problems as part of his work, his first love was physics. He strove for decades towards a unified field theory of general relativity and quantum mechanics. Lanczos began to apply the theoretical methods first developed by Hamilton to the problem of the unified field. Towards the end of his life, he published what he regarded as his greatest work in this area. It involved high frequency undulations in the metric, matter waves or “ripples on the pond” as his biographer Barbara Gellai called them.

The scientific writings of Lanczos are of great elegance and radiate his enthusiasm for mathematics and physics. He published more than 100 scientific papers and about 10 books. He was very conscious of the importance of beauty and elegance in mathematical and physical theories. His main goal was a unified theory that would embrace general relativity, electrodynamics and quantum mechanics. Although he did not reach this goal – and we have still not reached it – he made some profound contributions to our understanding of the universe. The quest for a unified theory of all the forces of nature is a problem of great difficulty and the struggle to achieve this continues.


Video Recordings: In the late 1960s, Lanczos gave a set of lectures at UMIST (now the University of Manchester). Video recordings of these lectures, in which Lanczos talks about mathematics, his life, and Einstein, are available here and here.

Biography of Lanczos: Gellai, Barbara, 2010: The Intrinsic Nature of Things: the Life and Science of Cornelius Lanczos. American Mathematical Society, 168pp.

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