Can a Matrix Have No Real Eigenvalues and Still Be Diagonalizable?

Introduction

When discussing the characteristics of matrices, one essential concept is eigenvalues. Eigenvalues play a crucial role in understanding the behavior of matrices, particularly in transformations and diagonalization. This article explores the intriguing question: can a matrix have no real eigenvalues and still be diagonalizable? We will also delve into the conditions under which this is possible, focusing on square matrices with complex coefficients.

Matrix Coefficients and Eigenvalues

A fundamental property to consider is that a matrix with coefficients in a non-algebraically closed field can have no real eigenvalues. This phenomenon is exemplified by a matrix with real coefficients that has no real eigenvalues but possesses complex eigenvalues in a different field of coefficients. Consider the following matrix:

[begin{pmatrix} -1 -1 2 1 end{pmatrix}]

This matrix, when considered with real coefficients, has no real eigenvalues. However, treating it as a matrix with complex coefficients, its eigenvalues are i and -i.

Square Matrices and Eigenvalues

A matrix that is not square cannot have eigenvalues or eigenvectors. On the other hand, a matrix that is square, by definition, must have at least one eigenvalue. This is due to the fact that its characteristic polynomial, which is the determinant of A - λI, is a polynomial of degree n when A is an n x n matrix. Any non-constant polynomial has at least one root in an algebraically closed field.

Example of a Matrix with Complex Eigenvalues

Consider the matrix A begin{pmatrix} 0 1 -1 0 end{pmatrix}. The characteristic polynomial of this matrix is λ^2 1, which has complex eigenvalues, specifically i and -i. Such a matrix can be diagonalized, resulting in a diagonal matrix with these complex eigenvalues on the main diagonal:

begin{pmatrix} i 0 0 -i end{pmatrix}

Rotation matrices often exhibit complex eigenvalues, making them an excellent example to study further in this context.

Diagnostic Measure of Determinant

To determine whether a square matrix has an eigenvalue, one can use the determinant as a diagnostic measure. If a matrix satisfies the equation A x c x, where c is a scalar and x is a column vector, then the column vector x must have the same number of rows as A has columns. Moreover, since the number of rows of x must be equal to the number of rows of A, it follows that A must be square.

Conditions for Every Square Matrix to Have an Eigenvalue

The answer to whether every square matrix has an eigenvalue depends on the scalar field used. In the field of complex numbers, every square matrix has an eigenvalue because this field is algebraically closed. However, in the field of real numbers, not every square matrix has an eigenvalue. For instance, skew-symmetric matrices of even order do not have real eigenvalues.

Quaternion Matrices and Eigenvalues

The concept of eigenvalues in matrices with quaternion entries is more complex. In such cases, both left and right eigenvalues are considered due to the non-commutative nature of quaternions. A quaternion matrix always has both left and right eigenvalues. The proof of the existence of left eigenvalues is topological in nature.

Understanding these concepts is crucial for anyone dealing with advanced linear algebra and applications in fields such as physics, engineering, and data science. Exploring the intricacies of diagonalization and eigenvalues in different fields of coefficients will broaden one's perspective on matrix theory.