Died: 1949 in Baltimore, USA

John Williamson was born in The Manse, Kinross, where his father was the Presbyterian minister. He attended the Edinburgh Academy from 1912 to 1919. He sat the Leaving Certificate examinations passing Mathematics, Latin, Greek, French, and English at Higher grade. After having passed the Preliminary Examination, Williamson first matriculated at the University of Edinburgh in October 1919.

As an undergraduate at Edinburgh, Williamson studied Ordinary Mathematics, Natural Philosophy and Greek in session 1919-20. His performance was outstanding and he was awarded the Bruce of Grangehill Bursary in June 1920. In his second session he studied Intermediate Honours Mathematics, Dynamics, and General Physics. He also took the Intermediate Honours course on Heat in Spring 1921, the Intermediate Honours course on Electricity in Summer 1921, and Ordinary Philosophy. Again his performance was outstanding and he was awarded the Horsliehill Bursary in June 1921. In session 1921-22 he took Advanced Mathematics, Electricity I, Waves, Dynamics, Heat, and Electricity II. He graduated M.A. with First Class Honours in Mathematics and Natural Philosophy on 13 July 1922.

He was appointed as a Lecturer in Mathematics at the University of St Andrews but, on the award of a Commonwealth Scholarship, he went to Chicago in 1925. After studying with L E Dickson and E H Moore, he was awarded a Ph.D. by Chicago in 1927 for his thesis Conditions for Associativity of Division Algebras Connected with Non-Abelian Groups. He then returned to his lectureship at the University of St Andrews. Turnbull writes [1]:-

He brought back to Scotland the fruits of what Wedderburn had taken to America which had inspired Dickson and formed the central theme of his lecture course at Chicago. Williamson was probably the first to lecture in Scotland (1927) upon algebras and their arithmetics: his interest in this led to original contributions in the region of division algebras.

Williamson was elected a Fellow of the Royal Society of Edinburgh on 5 March 1928 having been proposed by Herbert Westren Turnbull, Sir Edmund Taylor Whittaker, Ralph Allan Sampson, and Charles Glover Barkla.

Williamson left St Andrews when he accepted a Associate Professorship in Mathematics at Johns Hopkins University in Baltimore, USA. The announcement in the January 1929 part of the American Mathematical Monthly reported:-

Dr John Williamson has been appointed associate in mathematics at Johns Hopkins University.

On 1 February 1924, when he was a Lecturer in Mathematics at the University of St Andrews, Williamson joined the Edinburgh Mathematical Society. He read the paper On the Linear Complexes belonging to the Invariant System of Three Quadrics to the meeting of the Society on Friday 11 January 1924. It was at this meeting that he was proposed for membership, which was confirmed at the next meeting. Williamson read the paper The irreducible covariants belonging to the concomitant system of three quadrics to the meeting of the Society in St Andrews on Saturday 14 June 1924. At the meeting on Friday 16 January 1925 a joint paper by Williamson and Turnbull On the general invariant theory of two quadratics in n variables was read.

His obituary in the Edinburgh Mathematical Notes is at THIS LINK.

Williamson married Lydia Stevenson, who had been a student at St Andrews. 
writes:-

His early death in 1949, due to heart failure as he was returning from church one Sunday evening, has taken away one who came to be regarded in America as a leading exponent of the theory of matrices. Throughout his life he was devoted to teaching and mathematical research, producing an impressive stream of papers, on invariants, algebras and matrices. The general trend in this programme was to pass from the ordinary field of complex numbers to more restricted and specialised fields.

Also [1]:-

1. H W Turnbull, John Williamson obituary, Edinburgh Mathematical Notes 38 (1952), 23-24.
2. Graduates in Arts, 1884-1925 (University of Edinburgh).
3. Graduates in Arts (University of Edinburgh).