The Cayley Hamilton Theorem

M 101 Module I Lecture 10 Cayley Hamilton Theorem Lessons Blendspace 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. in this section, direct proofs are presented. as the examples above show, obtaining the statement of the cayley–hamilton theorem for an n × n. The cayley hamilton theorem forms an important concept that is widely used in the proofs of many theorems in pure mathematics. some of the important applications of this theorem are listed below: the cayley hamilton theorem is used to define vital concepts in control theory such as the controllability of linear systems.

Cayley Hamilton Theorem Youtube Since p(d) = 0, we conclude that p(a) = 0. this completes the proof of the cayley hamilton theorem in this special case. step 2: to prove the cayley hamilton theorem in general, we use the fact that any matrix a cn n can be approximated by diagonalizable ma trices. more precisely, given any matrix a cn n, we can find a sequence of matrices {ak. The theorem allows a n to be articulated as a linear combination of the lower matrix powers of a. if the ring is a field, the cayley–hamilton theorem is equal to the declaration that the smallest polynomial of a square matrix divided by its characteristic polynomial. example of cayley hamilton theorem 1.) 1 x 1 matrices. Let a be a 3 × 3 real orthogonal matrix with det (a) = 1. (a) if − 1 √3i 2 is one of the eigenvalues of a, then find the all the eigenvalues of a. (b) let a100 = aa2 ba ci, where i is the 3 × 3 identity matrix. using the cayley hamilton theorem, determine a, b, c. (kyushu university). The cayley hamilton theorem produces an explicit polynomial relation satisfied by a given matrix. in particular, if m m is a matrix and p {m} (x) = \det (m xi) pm (x) = det(m −xi) is its characteristic polynomial, the cayley hamilton theorem states that p {m} (m) = 0 pm (m) = 0.