# The Cayley-Hamilton Theorem

## The Cayley-Hamilton Theorem

Summary

1. (The Cayley-Hamilton Theorem) If $p(t)$ is the characteristic polynomial for an $n\times n$ matrix $A$, then the matrix $p(A)$ is the $n \times n$ zero matrix.

### Example

Let $A=\begin{bmatrix} 1& 1 \\ 1& 3 \end{bmatrix}$. The characteristic polynomial $p(t)$ of $A$ is
\begin{align*}
p(t)&=\det(A-tI)=\begin{bmatrix}
1-t& 1 \\
1& 3-t
\end{bmatrix}
\\
&=t^2-4t+2.
\end{align*}

Then the Cayley-Hamilton theorem says that the matrix $p(A)=A^2-4A+2I$ is the $2\times 2$ zero matrix. In fact, we can directly check this:
\begin{align*}
p(A)&=A^2-4A+2I=\begin{bmatrix}
1& 1 \\
1& 3
\end{bmatrix}\begin{bmatrix}
1& 1 \\
1& 3
\end{bmatrix}-4\begin{bmatrix}
1& 1 \\
1& 3
\end{bmatrix}+2\begin{bmatrix}
1& 0\\
0& 1
\end{bmatrix}\6pt] &=\begin{bmatrix} 2& 4 \\ 4& 10 \end{bmatrix} +\begin{bmatrix} -4& -4 \\ -4& -12 \end{bmatrix} +\begin{bmatrix} 2& 0 \\ 0& 2 \end{bmatrix} =\begin{bmatrix} 0& 0 \\ 0& 0 \end{bmatrix}. \end{align*} =solution ### Problems 1. Let T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}. Calculate and simplify the expression -T^3+4T^2+5T-2I, where I is the 3\times 3 identity matrix. (The Ohio State University) 2. Find the inverse matrix of the matrix A=\begin{bmatrix} 1 & 1 & 2 \\ 9 &2 &0 \\ 5 & 0 & 3 \end{bmatrix} using the Cayley–Hamilton theorem. 3. Find the inverse matrix of the 3\times 3 matrix A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix} using the Cayley-Hamilton theorem. 4. Let \[A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}. Find the eigenvalues and the eigenvectors of the matrix
$B=A^4-3A^3+3A^2-2A+8E.$ (Nagoya University)

5. Let $A, B$ be complex $2\times 2$ matrices satisfying the relation $A=AB-BA$. Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

6. In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not.
(a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , \lambda=-i$, and $\lambda=0$.
(b) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , \lambda=-i$, and $\lambda=-1$.

7. Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$. Then for each positive integer $n$ find $a_n$ and $b_n$ such that $A^{n+1}=a_nA+b_nI$, where $I$ is the $2\times 2$ identity matrix.

8. Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation $A^{n} = b_n A + c_n I$, where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and then find a recursive relationship to find $b_n, c_n$ for every $n \geq 1$.
9. Let $A$ be an $n\times n$ complex matrix. Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as
$p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers. Let $C$ be the companion matrix of the polynomial $p(x)$ given by
$C=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}= [\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],$ where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by $\mathbf{a}=\begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_{n-1} \end{bmatrix}$. Then prove that the following two statements are equivalent.

(1) There exists a vector $\mathbf{v}\in \C^n$ such that
$\mathbf{v}, A\mathbf{v}, A^2\mathbf{v}, \dots, A^{n-1}\mathbf{v}$ form a basis of $\C^n$.
(2) There exists an invertible matrix $S$ such that $S^{-1}AS=C$.
(Namely, $A$ is similar to the companion matrix of its characteristic polynomial.)

10. Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$. Let $A \in V$ and consider the set
$S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}$ of $n^2$ elements. Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

11. Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.
(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.
(b) Let $A^{100}=aA^2+bA+cI$, where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.
(Kyushu University)

12. Let $A$ and $B$ be $2\times 2$ matrices such that $(AB)^2=O$, where $O$ is the $2\times 2$ zero matrix. Determine whether $(BA)^2$ must be $O$ as well. If so, prove it. If not, give a counter example.