# Thomas Muir (mathematician)

Sir Thomas Muir FRS[1] (25 August 1844 – 21 March 1934) was a Scottish mathematician, remembered as an authority on determinants.[2][3][4] He was born in Stonebyres in South Lanarkshire, and brought up in the small town of Biggar. At the University of Glasgow he changed his studies from classics to mathematics after advice from the future Lord Kelvin. After graduating he held positions at the University of St Andrews and the University of Glasgow. From 1874 to 1892 he taught at Glasgow High School. In 1882 he published Treatise on the theory of determinants; then in 1890 he published a History of determinants. In his 1882 work, Muir rediscovered an important lemma that was first proved by Cayley 35 years earlier:[5]

If B is a skew symmetric matrix, then its determinant is equal to the square of its Pfaffian:

${\displaystyle \det(\mathbf {B} )=\operatorname {pf} ^{2}(\mathbf {B} )\,}$

From 1892 he was in South Africa working in education, and then in administration at the University of the Cape. He was elected a Fellow of the Royal Society of Edinburgh and in 1900 a Fellow of the Royal Society.[6] He was installed Companion of the Order of St Michael and St George (CMG) in 1901 and knighted in the 1915 Birthday Honours.

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 theorem on relations between minors. In more abstract language, it is a general result on the equations defining Grassmannians as algebraic varieties.

## 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, 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

## References

1. ^ H. W. T. (1934). "Sir Thomas Muir. 1844-1934". Obituary Notices of Fellows of the Royal Society. 1 (3): 179. doi:10.1098/rsbm.1934.0002.
2. ^ "Sir Thomas Muir, C.M.G., F.R.S". Nature. 132 (3329): 271–270. 1933. doi:10.1038/132271a0.
3. ^ H. W. T. (1934). "Sir Thomas Muir, C.M.G., F.R.S". Nature. 134 (3386): 449–440. doi:10.1038/134449a0.
4. ^
5. ^ Cayley, Arthur (1847). "Sur les determinants gauches" [On skew determinants]. Crelle's Journal. 38: 93–96.
6. ^ "Library and Archive Cataloge". Royal Society. Retrieved 2012-03-05.