Web1 The Cayley-Hamilton theorem The Cayley-Hamilton theorem Let A ∈Fn×n be a matrix, and let p A(λ) = λn + a n−1λn−1 + ···+ a 1λ+ a 0 be its characteristic polynomial. Then An + a n−1An−1 + ···+ a 1A+ a 0I n = O n×n. The Cayley-Hamilton theorem essentially states that every square matrix is a root of its own characteristic polynomial. WebFeb 10, 2015 · $\begingroup$ @Blah: Here is the more relevant subpage of the wiki article. The main point is that the proposed proof want to boil down to computing (just) the determinant of a zero matrix, and none of the formal tricks can justify that.
A Generalization of the Cayley-Hamilton Theorem - ResearchGate
WebA matrix A∈Fn×nis diagonalizable if it is similar to some diagonal matrix in Fn×n. To diagonalize a matrix A∈Fn×nmeans to find a diagonal matrix Dand an invertible matrix P, both in Fn×n, such that D= P−1AP. Theorem 4.2. A matrix A∈Fn×n is diagonalizable if and only if Fn has a basis formed by eigenvectors of A. Proof. Fix a matrix ... WebFeb 26, 2016 · The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own characteristic polynomial. offshore angola oil fields map
matrices - Calculate $A^5 - 27A^3 + 65A^2$, where $A$ is the matrix …
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. If A is a given n × n … See more Determinant and inverse matrix For a general n × n invertible matrix A, i.e., one with nonzero determinant, A can thus be written as an (n − 1)-th order polynomial expression in A: As indicated, the Cayley–Hamilton … See more The Cayley–Hamilton theorem is an immediate consequence of the existence of the Jordan normal form for matrices over algebraically closed fields, see Jordan normal form § Cayley–Hamilton theorem See more • Companion matrix See more • "Cayley–Hamilton theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • A proof from PlanetMath. See more The above proofs show that the Cayley–Hamilton theorem holds for matrices with entries in any commutative ring R, and that … See more 1. ^ Crilly 1998 2. ^ Cayley 1858, pp. 17–37 3. ^ Cayley 1889, pp. 475–496 4. ^ Hamilton 1864a See more WebCayley-Hamilton theorem if p(s) = a0 +a1s+···+aksk is a polynomial and A ∈ Rn×n, we define p(A) = a0I +a1A+···+akAk Cayley-Hamilton theorem: for any A ∈ Rn×n we have X(A) = 0, where X(s) = det(sI −A) example: with A = 1 2 3 4 we have X(s) = s2 −5s−2, so X(A) = A2 −5A−2I = 7 10 15 22 −5 1 2 3 4 −2I = 0 Jordan canonical ... WebCayley Hamilton Theorem Let A A be a 2×2 2 × 2 matrix and let pA(λ) =λ2 +aλ+b p A ( λ) = λ 2 + a λ + b be the characteristic polynomial of A A. Then pA(A)= A2 +aA+bI2 = 0. p A ( A) = A 2 + a A + b I 2 = 0. Proof Suppose B =P−1AP B = P − 1 A P and A A are similar matrices. We claim that if pA(A) =0 p A ( A) = 0, then pB(B) = 0 p B ( B) = 0. offshore angler t shirts