# An Example of a Real Matrix that Does Not Have Real Eigenvalues

## Problem 596

Let
$A=\begin{bmatrix} a & b\\ -b& a \end{bmatrix}$ be a $2\times 2$ matrix, where $a, b$ are real numbers.
Suppose that $b\neq 0$.

Prove that the matrix $A$ does not have real eigenvalues.

## Proof.

Let $\lambda$ be an arbitrary eigenvalue of $A$.
Then the matrix $A-\lambda I$ is singular, where $I$ is the $2\times 2$ identity matrix.
This is equivalent to having $\det(A-\lambda I)=0$.

We compute the determinant as follows.
We have
\begin{align*}
\det(A-\lambda I)&=\begin{vmatrix}
a-\lambda & b\\
-b& a-\lambda
\end{vmatrix}\6pt] &=(a-\lambda)^2-b(-b)\\ &=a^2-2a\lambda+\lambda^2+b^2\\ &=\lambda^2-2a\lambda+a^2+b^2. \end{align*} We solve the equation \lambda^2-2a\lambda+a^2+b^2=0 by the quadratic formula and obtain \begin{align*} \lambda &=\frac{2a\pm\sqrt{4a^2-4(a^2+b^2)}}{2}=\frac{2a\pm\sqrt{-4b^2}}{2}\\[6pt] &=a\pm |b|i. \end{align*} Since b\neq 0 by assumption, the eigenvalue \lambda=a\pm|b|i is not a real number. As \lambda is an arbitrary eigenvalue of A, we conclude that all eigenvalues of A are not real numbers. ### More from my site • Common Eigenvector of Two Matrices and Determinant of Commutator Let A and B be n\times n matrices. Suppose that these matrices have a common eigenvector \mathbf{x}. Show that \det(AB-BA)=0. Steps. Write down eigenequations of A and B with the eigenvector \mathbf{x}. Show that AB-BA is singular. A matrix is […] • 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 […] • Use the Cayley-Hamilton Theorem to Compute the Power A^{100} 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 […]
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