Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.
The matrix is called the companion matrix of the polynomial $p(x)$.
Use the mathematical induction. The base case is clear.
For the induction step, use the cofactor expansion and apply the induction hypothesis.
Proof.
We prove that $p(x)=\det(xI-A)$ by induction on $n$.
The base case $n=1$ is clear since $A=[-a_0]$ is a $1 \times 1$ matrix and $\det(xI-A)=\det[x+a_0]=x+a_0$.
Induction step is as follows. Suppose that we have $p(t)=\det(xI-A)$ is true for a degree $n-1$ polynomial $p(t)$ and its companion matrix $A$.
We prove the statement for a degree $n$ polynomial.
Use the cofactor expansion corresponding to the first row, we obtain
\begin{align*}
\det(xI-A)&=\begin{vmatrix}
x & 0 & \dots & 0 &a_0 \\
-1 & x & \dots & 0 & a_1 \\
0 & -1 & \dots & 0 & a_2 \\
\vdots & & \ddots & & \vdots \\
0 & 0 & \dots & -1 & x+a_{n-1}
\end{vmatrix}\\[8pt]
&=x \begin{vmatrix}
x & 0 & \dots & 0 &a_1 \\
-1 & x & \dots & 0 & a_2 \\
\vdots & & \ddots & & \vdots \\
0 & 0 & \dots & 1 & x+a_{n-1}
\end{vmatrix}
+(-1)^{n+1}a_0
\begin{vmatrix}
-1 & x & \dots & 0 \\
0 & -1 & \dots & 0 \\
\vdots & & \ddots & \vdots \\
0 & 0 & \dots & -1
\end{vmatrix}.
\end{align*}
Now by the induction hypothesis, the first determinant is
\[x^{n-1}+a_{n-1}x^{n-2}+\cdots+a_2 x+a_1.\]
The second determinant is $(-1)^{n-1}$ since it is an $(n-1)\times (n-1)$ triangular matrix, determinant is the product of diagonal entries.
Therefore we have
\begin{align*}
\det(xI-A)&=x(x^{n-1}+a_{n-1}x^{n-2}+\cdots+a_2x+a_1)+(-1)^{n+1}a_0(-1)^{n-1}\\
&=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=p(x).
\end{align*}
Thus the statement is true for a degree $n$ polynomial. By induction, we complete the proof.
Algebraic Number is an Eigenvalue of Matrix with Rational Entries
A complex number $z$ is called algebraic number (respectively, algebraic integer) if $z$ is a root of a monic polynomial with rational (respectively, integer) coefficients.
Prove that $z \in \C$ is an algebraic number (resp. algebraic integer) if and only if $z$ is an eigenvalue of […]
Rotation Matrix in Space and its Determinant and Eigenvalues
For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an […]
Maximize the Dimension of the Null Space of $A-aI$
Let
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
\end{bmatrix}.\]
Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
Your score of this problem is equal to that […]
Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable
Determine all eigenvalues and their algebraic multiplicities of the matrix
\[A=\begin{bmatrix}
1 & a & 1 \\
a &1 &a \\
1 & a & 1
\end{bmatrix},\]
where $a$ is a real number.
Proof.
To find eigenvalues we first compute the characteristic polynomial of the […]
Find the Inverse Matrix Using the Cayley-Hamilton Theorem 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.
Solution.
To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial $p(t)$ of […]
Find All the Eigenvalues of Power of Matrix and Inverse Matrix
Let
\[A=\begin{bmatrix}
3 & -12 & 4 \\
-1 &0 &-2 \\
-1 & 5 & -1
\end{bmatrix}.\]
Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.
Proof.
We first determine all the eigenvalues of the matrix […]
2 Responses
[…] For a basic question about the companion matrix of a polynomial, check out the post “Companion matrix for a polynomial“. […]
[…] post Companion matrix for a polynomial for the definition of the companion matrix and the proof of the above […]