## These guys shapes the modern matrix theory

I copied this post from Dan Simon’s book “Optimal State Estimation”, section 1.1.4. It’s interesting to see that most of the main framework of matrix theory was built in relative recent time.

In spite of this very early beginning (the use of matrices by Babylonians and ancient Chinese) it was not until the end of 17th century that serious investigation of matrix algebra began. In 1683, the Japanese mathematician Takakazu Seki Kowa wrote a book called “Method of Solving the Dissimulated Problems”. This book gives general methods for calculating determinants and presents examples for matrices as large as 5×5. Coincidentally, in the same year (1683) Gottfried Leibniz in Europe also first used determinants to solve systems of linear equations. Leibniz also discovered that a determinant could be expanded using any of the matrix columns.

In the middle of the 1700s, Colin Maclaurin and Gabriel Cramer published some major contributions to matrix theory. After that point, work on matrices became rather regular, with significant contributions by Etienne Bezout, Alexandre Vandermonde, Pierre Laplace, Joseph Lagrange, and Carl Gauss. The term “determinant” was first used in the modern sense by Augustin Cauchy in 1812 (although the word was used earlier by Gauss in a different sense). Cauchy also discovered matrix eigenvalues and diagonalization, and introduced the idea of similar matrices. He was the first to prove that every real symmetric matrix is diagonalizable.

James Sylvester (in 1850) was the first to use the term “matrix”. Sylvester moved to England in 1851 to became a lawyer and met Arthur Cayley, a fellow lawyer who was also interested in mathematics. Cayley saw the importance of the idea of matrices and in 1853 he invented matrix inversion. Cayley also proved that 2×2 and 3×3 matrices satisfy their own characteristic equations. The facet that a matrix satisfies its own characterization equation is now called Cayley-Hamilton theorem. The theorem has William Hamilton’s name associated with it because he proved the theorem for 4×4 matrices during the course of his work on quaternions.

Camille Jordan invented the Jordan canonical form of a matrix in 1870. Georg Frobenius proved in 1878 that all matrices satisfy their own characteristic equation (the Cayley Hamilton theorem). He also introduced the definition of the rank of a matrix. The nullity of a square matrix was defined by Sylvester in 1884. Karl Weierstrass’s and Leopold Kronecker’s publication in 1903 were instrumental in establishing matrix theory as an important branch of mathematics. Leon Mirsky’s book in 1955 helped solidify matrix theory as a fundamentally important topic in university mathematics.

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