Questions askedCategory: 2568Eigenvalues and the Characteristics Polynomial

Hello Sir, I am your student, Anthony from MATH 2568. May I know what is Cayley-Hamilton Theorem? I don\’t remember it was taught in class, and I cannot find what is it in notes either.  Hope to hear back from you soon. Thank you.   Regards, Anthony

Yu Staff answered 7 months ago

We didn’t cover Cayley-Hamilton Theorem in class. If we have time after finishing all sections, we might come back and study Cayley-Hamilton Theorem in class.

Anyway, here is the ideal. Let me use a specific example. Consider the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ 1& 2 \end{bmatrix}$. The characteristic polynomial of $A$ is $p(t)=\det(A-tI)=t^2-4t+5$.

Then “substitute” the matrix $A$ into its characteristic polynomial $p(t)$. Then we have
$p(A)=A^2-4A+5I.$

Note that you cannot naively substitute $A$ in $p(t)$. You need to make the constant term $5$ to be $5I$.

Then Cayley-Hamilton theorem tells you that $p(A)=A^2-4A+5I$ is always the zero matrix. (Please compute this directly and check.)

In general, the statement of Cayley-Hamilton theorem is as follows.
Let $A$ be an $n\times n$ matrix and let $p(t)$ be its characteristic polynomial. Then the matrix $p(A)$ obtained by substituting $A$ into $p(t)$ is always the zero matrix.

With this in mind, try to do exercise problems in section 4.4.

Thank you, Sir. Then, could I say that A^2, A, and I are vectors, and 1, -4, 5 are coefficients? The set { A^2, A, I } is linearly dependent?

Yu Staff replied 7 months ago

If you consider these matrices as vectors in the vector space of all $2\times 2$ matrices, then the answer is yes.