chapter  4
94 Pages

Eigenvalues, Eigenvectors, and Diagonalization

There are various important scalars associated with a square matrix, numbers which are somehow characteristic of the nature of the matrix and which simplify calculations with the matrix. One of the most useful of these scalars is the determinant. The concept of a determinant has an ancient and honorable history strewn with the names of many prominent mathematicians and scientists. Its ‘‘modern’’ development begins in the late seventeenth century with the independent work of the Japanese mathematician Seki Kowa (1642-1708) and of Gottfried Wilhelm Leibniz (1646-1716), the coinventor of calculus. At one time, determinants were the most important tool used to analyze and solve systems of linear equations, while matrix theory played only a supporting role. Currently, there are various schools of thought on the importance of determinants in modern mathematical and scientific work. As one author* has described the situation:

. . . mathematics, like a river, is everchanging in its course, and major branches can dry up to become minor tributaries while small trickling brooks can develop into raging torrents. This is precisely what occurred with determinants and matrices. The study and use of determinants eventually gave way to Cayley’s matrix algebra, and today matrix and linear algebra are in the main stream of applied mathematics, while the role of determinants has been relegated to a minor backwater position.