Thomas Muir (mathematician)
Encyclopedia
Sir Thomas Muir FRS
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

 (25 August 1844 – 21 March 1934) was a Scottish
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, remembered as an authority on determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

s. He was born in Stonebyres in South Lanarkshire
South Lanarkshire
South Lanarkshire is one of 32 unitary council areas of Scotland, covering the southern part of the former county of Lanarkshire. It borders the south-east of the city of Glasgow and contains many of Glasgow's suburbs, commuter towns and smaller villages....

, and brought up in the small town of Biggar. At the University of Glasgow
University of Glasgow
The University of Glasgow is the fourth-oldest university in the English-speaking world and one of Scotland's four ancient universities. Located in Glasgow, the university was founded in 1451 and is presently one of seventeen British higher education institutions ranked amongst the top 100 of the...

 he changed his studies from classics to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 after advice from the future Lord Kelvin. After graduating he held positions at the University of St Andrews
University of St Andrews
The University of St Andrews, informally referred to as "St Andrews", is the oldest university in Scotland and the third oldest in the English-speaking world after Oxford and Cambridge. The university is situated in the town of St Andrews, Fife, on the east coast of Scotland. It was founded between...

 and the University of Glasgow
University of Glasgow
The University of Glasgow is the fourth-oldest university in the English-speaking world and one of Scotland's four ancient universities. Located in Glasgow, the university was founded in 1451 and is presently one of seventeen British higher education institutions ranked amongst the top 100 of the...

. From 1874 to 1892 he taught at Glasgow High School
High School of Glasgow
The High School of Glasgow is an independent, co-educational day school in Glasgow, Scotland. Founded as the Choir School of Glasgow Cathedral in around 1124, it is the oldest school in Scotland, and the twelfth oldest in the United Kingdom. It remained part of the Church as the city's grammar...

. In 1882 he published Treatise on the theory of determinants; then in 1890 he published a History of determinants. In his 1882 work, Muir proved an important lemma:

If B is a skew symmetric matrix, then its determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 is equal to its Pfaffian
Pfaffian
In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. This polynomial is called the Pfaffian of the matrix, The term Pfaffian was introduced by who named them after Johann Friedrich Pfaff...

 squared:


From 1892 he was in South Africa
South Africa
The Republic of South Africa is a country in southern Africa. Located at the southern tip of Africa, it is divided into nine provinces, with of coastline on the Atlantic and Indian oceans...

 working in education, and then in administration at the University of the Cape. He was knighted in 1910.

From 1906 onwards he published a five-volume expansion of his history of determinants, the final part (1929) taking the theory to 1920. A further book followed in 1930.

His name now attaches to a duality
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...

 theorem on relations between minors. In more abstract language, it is a general result on the equations defining Grassmannian
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

s as algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

.

Publications by Sir Thomas Muir

  • The Theory of the Determinant in the Historical Order of Development. 4 vols. New York: Dover Publications 1960
  • A Treatise on the Theory of Determinants. Revised and Enlarged by William H. Metzler. New York: Dover Publications 1960
  • "A Second Budget of Exercises on Determinants", American Mathematical Monthly
    American Mathematical Monthly
    The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....

    , Vol. 31, No. 6. (June, 1924), pp. 264–274
  • "Note on the Transformation of a Determinant into any Other Equivalent Determinant", The Analyst, Vol. 10, No. 1. (Jan 1883), pp. 8–9

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