#DEB [Arthur Cayley](http://wikipedia.org/en/Arthur_Cayley) was a renowned British mathematician and lawyer who lived from 1821 to 1895. He made significant contributions to various branches of mathematics, such as algebra, geometry, and matrix theory. Cayley was born in Richmond, England, and showed an early aptitude for mathematics. He attended Cambridge University, where he excelled in his studies and graduated as a Senior Wrangler (the top-ranked mathematics student) in 1842. He then pursued a legal career but continued to pursue his passion for mathematics alongside his legal practice. In the field of algebra, Cayley made numerous important discoveries. He introduced the concept of matrices, laying the foundation for modern matrix theory. His work on matrices led to advancements in linear algebra and its applications in physics and engineering. Cayley also developed the theory of algebraic invariants and contributed significantly to group theory. In geometry, Cayley made substantial contributions to projective geometry and introduced the concept of homogeneous coordinates. He utilized these coordinates to generalize Euclidean geometry and solve geometric problems more elegantly. His work on geometric transformations laid the groundwork for modern transformational geometry. Cayley's mathematical achievements were extensive, with over 900 published papers throughout his career. His work covered diverse areas like calculus, number theory, complex analysis, logic, probability theory, and more. Despite being primarily self-taught in mathematics due to his legal profession, Cayley became a prominent figure within the mathematical community. He became President of the London Mathematical Society twice and received numerous honors during his lifetime for his contributions to mathematics. Arthur Cayley's work has had a lasting impact on various branches of mathematics. His ideas continue to be studied and applied by mathematicians worldwide. He is recognized as one of the most influential mathematicians of the 19th century and remains highly regarded for his significant contributions to the field. # What is the relation between Arthur Cayley and William Rowan Hamilton Arthur Cayley and [[William Rowan Hamilton]] were both prominent mathematicians in the 19th century. While they did not have a direct personal relationship, their work and contributions to the field of mathematics intersected in several ways. One of the notable connections between Cayley and Hamilton is their work on matrix theory. Cayley made significant contributions to the development of matrix algebra and is often referred to as the "father of modern matrix theory." He introduced many fundamental concepts in linear algebra, including the notion of a matrix as an array of numbers. Hamilton, on the other hand, is known for his development of quaternions, which are a type of mathematical object that extends complex numbers. Quaternions have important applications in various fields, including computer graphics and robotics. In his study of quaternions, Hamilton used matrices extensively and developed methods for representing them. Cayley also made contributions to [[Quaternions|quaternions]], particularly in establishing their connection to matrices. He showed that every quaternion could be represented by a 2x2 matrix with complex entries. This work further bridged the gap between quaternions and matrices. In addition to their work on matrices and quaternions, both Cayley and Hamilton made significant contributions to other areas of mathematics. Cayley was known for his work on group theory, algebraic geometry, and invariant theory. Hamilton made important contributions to calculus, optics, dynamics, and number theory. Although they may not have collaborated directly or had a personal relationship, Arthur Cayley and William Rowan Hamilton were two influential mathematicians whose work intersected in several areas of mathematics. Their contributions continue to have a lasting impact on the field today. # Cayley-Hamiliton theorem The Cayley-Hamilton theorem is a fundamental result in linear algebra named after the mathematicians Arthur Cayley and [[Literature/People/William Rowan Hamilton|William Rowan Hamilton]]. It states that every square matrix satisfies its own characteristic equation. In other words, if A is an n × n matrix with entries in a field, then the characteristic polynomial of A, denoted by p(λ), is given by p(λ) = det(A - λI), where I is the identity matrix of size n. The Cayley-Hamilton theorem asserts that if we substitute the matrix A into its own characteristic polynomial, i.e., evaluate p(A), then the resulting expression will be the zero matrix. Mathematically, this can be expressed as: p(A) = 0, where 0 represents the zero matrix of size n. The significance of this theorem lies in its application to various areas of mathematics and engineering. It allows us to find powers and functions of matrices easily by replacing them with their characteristic polynomials. It also provides a powerful tool for solving systems of linear differential equations and studying properties of linear operators. Overall, the Cayley-Hamilton theorem provides a deep insight into the relationship between matrices and their characteristic polynomials, paving the way for many important results in linear algebra and related fields. # References ```dataview TABLE title as TITLE, year as YEAR FROM "Literature/Reading notes" Where contains(authors, "Arthur Cayley") or contains(subject, "Arthur Cayley") SORT file.mtime asc ```