Cayley Hamilton Theorem

Simple expalin some examples and daily use

1. Cayley–Hamilton Theorem What is Cayley Hamilton Theorem ? Who found it ? Where we use it ?

2. About Cayley - Hamilton Arthur Cayley :(1821 – 1895) was a British Mathematician.He helped found the modern British school of pure mathematics.As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3 He was the first to define the concept of a group in the modern way as a set with a binary operation satisfying certain laws. Formerly , when mathematicians spoke of "groups", they had meant permutation groups. Cayley's theorem is named in honour of Cayley.

3. About Sir William Rowan Hamilton Sir William Rowan Hamilton: (1805 -1865) was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques. His best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions. Hamilton is said to have shown immense talent at a very early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton, 'This young man, I do not say will be, but is, the first mathematician of his age.'

4. Cayley-Hamilton Theorem In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation. If A is a given n×n matrix and In is the n×n identity matrix, then the characteristic polynomial of A is defined as where det is the determinant operation and λ is a scalar element of the base ring. Since the entries of the matrix are (linear or constant) polynomials in λ, the determinant is also an n-th order monic polynomial in λ. The Cayley–Hamilton theorem states that substituting the matrix A for λ in this polynomial results in the zero matrix,

5. The powers of A, obtained by substitution from powers of λ, are defined by repeated matrix multiplication; the constant term of p(λ) gives a multiple of the power A , which power is defined as the identity matrix. The theorem allows An to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. The theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton. This corresponds to the special case of certain real 4 × 4 real or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858 stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. The general case was first proved by Frobenius in 1878. Cayley-Hamilton Theorem

6. Simple Examples 1×1 matrices 2×2 matrices As a concrete example, let Its characteristic polynomial is given by The Cayley–Hamilton theorem claims that, if we define We can verify by computation that indeed,

7. A Direct Algebraic Proof This proof uses just the kind of objects needed to formulate the Cayley–Hamilton theorem: matrices with polynomials as entries. The matrix t In −A whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials form a commutative ring, it has an adjugate Then, according to the right-hand fundamental relation of the adjugate, one has Since B is also a matrix with polynomials in t as entries, one can, for each i , collect the coefficients of ti in each entry to form a matrix B i of numbers, such that one has While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t i has been written to the left of the matrix to stress this point of view.

8. Cayley - Hamilton theorem is useful to find 1.Inverse & power of matrix 2.Only inverse of matrix. 3.Only power of matrix.

9. Uses of Cayley-Hamilton Theorem in Daily Life Matrices are used in many fields like robotics, automation, encryption, quantum mechanics, electrical circuits, 3D visualization in 2D etc. All places you would require powers of the matrix or inverse. Cayley Hamilton's theorem helps in expressing the inverse as a polynomial expression of the matrix. Higher powers of the matrix in terms of the lower lower powers of the matrix . The Cayley -Hamilton theorem and its generalizations have been used in control systems, electrical circuits, systems with delays, singular systems. specially in DC circuit

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