Cayley–Hamilton theorem - Wikipedia
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 numbers or the integers) satisfies its own characteristic equation.
Cayley Hamilton Theorem - Statement, Formula, Proof, Examples
Cayley Hamilton Theorem states that a square matrix (real or complex) will satisfy its own characteristic polynomial equation. Understand Cayley Hamilton Theorem using solved examples.
The Cayley-Hamilton theorem Theorem 1. Let A be a n × n matrix, and let p(λ) = det(λI − A) be the characteristic polynomial of A. Then p(A) = 0. Proof. Step 1: Assume first that A is diagonalizable. In this case, we can find an invertible matrix S and a diagonal matrix D = λ 1 0 ... 0 0... ..... 0 0 ... 0 λ n
Cayley-Hamilton Theorem | Statement & Proof | Examples - BYJU'S
In linear algebra, the Cayley–Hamilton theorem (termed after the mathematicians Arthur Cayley and William Rowan Hamilton) says that every square matrix over a commutative ring (for instance the real or complex field) satisfies its own characteristic equation.
We present three proofs for the Cayley-Hamilton Theorem. The nal proof is a corollary of the Jordan Normal Form Theorem, which will also be proved here. 1. Introduction. The Cayley-Hamilton Theorem states that any square matrix satis es its own characteristic polynomial.
Cayley Hamilton Theorem: Statement, Formula, Proof, and …
2024年5月30日 · What is Cayley Hamilton's Theorem? The Cayley Hamilton Theorem says that for any square matrix (which has the same number of rows and columns), the matrix will always follow a certain equation. This equation is called the characteristic polynomial.
Cayley-Hamilton Theorem 1 (Cayley-Hamilton) A square matrix A satisfies its own characteristic equation. If p(r) = ( r)n + a n 1( r) n 1 + a 0, then the result is the equation ( nA) + a n 1( A)n 1 + + a 1( A) + a 0I = 0; where I is the n n identity matrix and 0 is the n n zero matrix.
Cayley-Hamilton Theorem | Brilliant Math & Science Wiki
The Cayley-Hamilton theorem produces an explicit polynomial relation satisfied by a given matrix. In particular, if \(M\) is a matrix and \(p_{M} (x) = \det(M-xI)\) is its characteristic polynomial, the Cayley-Hamilton theorem states that \(p_{M} (M) = 0\).
Cayley-Hamilton theorem - Statlect
The Cayley-Hamilton theorem shows that the characteristic polynomial of a square matrix is identically equal to zero when it is transformed into a polynomial in the matrix itself. In other words, a square matrix satisfies its own characteristic equation.
the Cayley-Hamilton theorem. Theorem 2.1. For any = (a ij) 2Mat n(A) with n 1 and its characteristic polynomial ˜= xn+ 2 A[x], the matrix ˜( ) 2Mat n(A) vanishes. To prove this result, we rst note that when Ais a eld then this is the usual Cayley-Hamilton theorem. We will reduce the general case to the special case of elds, or even just ...