Answer by egreg for Using the converse if Cayley Hamilton theorem
The Cayley-Hamilton theorem has nothing to do with this business.If $\lambda$ is an eigenvalue of $M$, then $Mv=\lambda v$, for some $v\ne0$. It follows that$$(M^2-5M+6I)v=(\lambda^2-5\lambda+6)v$$and...
View ArticleAnswer by Disintegrating By Parts for Using the converse if Cayley Hamilton...
Suppose that $p(M)=0$ for some square matrix $M$ and some polynomial$$ p(\lambda)=\lambda^k+a_{k-1}\lambda^{k-1}+\cdots + a_{1}\lambda+a_0.$$Then$$ p(M)-p(\lambda)I = -p(\lambda)I.$$You can rewrite the...
View ArticleAnswer by Federico for Using the converse if Cayley Hamilton theorem
What the author is doing is using that if a matrix $M$ satisfies a polynomial $p(t)$, the minimal polynomial of $M$ divides $p(t)$. As all the eigenvalues of $M$ appear as roots of the minimal...
View ArticleUsing the converse if Cayley Hamilton theorem
My textbook says: Let $M$ be a $3 \times 3$ Hermitian matrix which satisfies the matrix equation$$M^{2}-5 M+6 I=0$$Where $I$ refers to the identity matrix. Which of the following are possible...
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