Linear Algebra

There is at Least One Real Eigenvalue of an Odd Real Matrix

Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra

Problem 407

Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.
Prove that the matrix $A$ has at least one real eigenvalue.

Add to solve later

Proof 1.

Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$.
It is a degree $n$ polynomial and the coefficients are real numbers since $A$ is a real matrix. Since $n$ is odd, the leading term of $p(t)$ is $-t^n$. That is, we have
\[p(t)=-t^n+\text{lower terms}.\] (Note: if you use the alternative definition of characteristic polynomial $p(t)=\det(tI-A)$, then the leading term is $t^n$. You can easily modify the proof for this case.)

Therefore, as $t$ increases the polynomial $p(t)$ tends to $-\infty$:
\[\lim_{t\to \infty}=-\infty.\] Similarly, we have
\[\lim_{t \to -\infty}=\infty.\] By the intermediate value theorem, there is $-\infty < \lambda < \infty$ such that$p(\lambda)=0$. Since a root of $p(t)$ is an eigenvalue of $A$, we have obtained a real eigenvalue $\lambda$ of $A$.

Proof 2.

Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$ and write it as
\[p(t)=\prod_{i=1}^k(\lambda_i-t)^{n_i},\] where $\lambda_i$ are distinct eigenvalues of $A$ and $n_i$ is the algebraic multiplicity of $\lambda_i$.

Since $A$ is a real matrix, the coefficients of $p(t)$ are real numbers. So we have
\[\overline{p(t)}=p(\,\bar{t}\,).\] Thus, we have
\begin{align*}
p(\bar{t})&=\overline{p(t)}=\prod_{i=1}^k(\,\bar{\lambda}_i-\bar{t}\,)^{n_i}.
\end{align*}
Replacing $\bar{t}$ by $t$, we obtain
\begin{align*}
p(t)=\prod_{i=1}^k(\bar{\lambda}_i-t)^{n_i}.
\end{align*}
This yields that the complex conjugate $\bar{\lambda}_i$ is an eigenvalue with algebraic multiplicity $n_i$.
Hence each non-real eigenvalue $\lambda$ appears as a pair $(\lambda, \bar{\lambda})$ counting multiplicity.
Thus, there are even number of non-real eigenvalues of $A$.

The number of all eigenvalues of $A$ counting multiplicities is $n$.
Since $n$ is odd, there must be at least one real eigenvalue of $A$.

Add to solve later

More from my site

Complex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues Let $A$ be an $n\times n$ real matrix. Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$. We give two proofs. Proof 1. Let $\mathbf{x}$ be an eigenvector corresponding to the […]
How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]
Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities (a) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.\] Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue. (b) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 […]
$Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials$ Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials Let $A$ be a $2\times 2$ real symmetric matrix. Prove that all the eigenvalues of $A$ are real numbers by considering the characteristic polynomial of $A$. Proof. Let $A=\begin{bmatrix} a& b \\ c& d \end{bmatrix}$. Then […]
Eigenvalues and Eigenvectors of Matrix Whose Diagonal Entries are 3 and 9 Elsewhere Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 3 & 9 & 9 & 9 \\ 9 &3 & 9 & 9 \\ 9 & 9 & 3 & 9 \\ 9 & 9 & 9 & 3 \end{bmatrix}.\] (Harvard University, Linear Algebra Final Exam Problem) Hint. Instead of […]
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 of a Hermitian Matrix are Real Numbers Show that eigenvalues of a Hermitian matrix $A$ are real numbers. (The Ohio State University Linear Algebra Exam Problem) We give two proofs. These two proofs are essentially the same. The second proof is a bit simpler and concise compared to the first one. […]