Cayley hamilton theorem pdf

# Cayley hamilton theorem pdf
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StepTo prove the Cayley-Hamilton theorem in general, we use the fact that any matrix A Cn n can be approximated by diagonalizable ma-trices. Then cT(T) =Equivalently: If A Computing the Matrix Exponential The Cayley-Hamilton MethodThe matrix exponential eAt forms the basis for the homogeneous (unforced) and the forced response of LTI The Cayley-Hamilton Theorem. The purpose of this note is to give an elementary proof of the following result: Theorem. Recall the usual notationsR is a commutative ring with 1RM is an R-free module, A = (1; ; m) and R-basis of MRm m and EndR(M) the corresponding R-algebrasWe set R ~:= R[t] the ring of polynomials in t over R Cayley-Hamilton Theorem(Cayley-Hamilton) A square matrix Asatisﬁes its own characteristic equation. If p(r) = (r)n+ a n 1(r) n+ a 0, then the result is the equation (nA) + a n 1(A)n+ + a 1(A) + a 0I= 0; where Iis the n nidentity matrix andis the n nzero matrix More precisely, given any matrix A Cn n, we can find a sequence of matrices {Ak Cayley Hamilton theorem. MTHSC Section { Cayley’s Theorem Kevin James Kevin James MTHSC Section { Cayley’s Theorem. One finds that the characteristic e quation of A is det (A − λI) = − λ+ λ+ λ –=The matrix A is invertible because a= −≠By the Cayley-Hamilton theorem –A3 + A2 + A – I =or A(– A2 + A + I) = I. or A. −= – A+ A + I =− The Cayley– Hamilton Theorem asserts that if one substitutes A for λ in this polynomial, then one obtains the zero matrix. n matrix with entries in Math The Cayley{Hamilton Theorem Recall the usual notationsRis a commutative ring withRM is an R-free module, A= (1;; m) and R-basis of MR m and End Since p(D) = 0, we conclude that p(A) =This completes the proof of the Cayley-Hamilton theorem in this special case. This result is true for any square matrix with entries in a commutative ring. ∗Written for the course Mathematics at Brooklyn College of CUNY Math The Cayley{Hamilton Theorem. Theorem Every group is isomorphic to a group of The Cayley–Hamilton Theorem Theorem. Let V be a ﬁnite-dimensional vector space over a ﬁeld F, and let T: V → V be a linear transformation. (Cayley-Hamilton) Let A be an n.