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. 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.

  7. 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.)

  8. 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 $A\in V$.

  9. 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)

  10. 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.