Non-Similar Matrices with Identical Characteristic and Minimal Polynomials: A Curious Insight

Can two 5x5 matrices share the same characteristic and minimal polynomials but fail to be similar? The answer is a resounding yes. This article explores an intriguing example involving such matrices, shedding light on the concepts of characteristic polynomial, minimal polynomial, and the notion of matrix similarity.

Introduction

The study of matrices is fundamental in linear algebra, and understanding their properties such as characteristic and minimal polynomials is crucial. Additionally, the concept of similarity between matrices is essential in determining their equivalence. However, two matrices can possess the same characteristic and minimal polynomials but not be similar, as illustrated by the example provided in this article.

Characteristics of Matrices

The characteristics of interest in this context include the characteristic polynomial and the minimal polynomial. The characteristic polynomial is derived from the determinant of a matrix, specifically from the determinant of (A - lambda I), where (A) is the matrix, (lambda) is an eigenvalue, and (I) is the identity matrix. The minimal polynomial is the monic polynomial of the lowest degree such that (m(A) 0). Importantly, the minimal polynomial divides the characteristic polynomial.

Non-Similar Matrices with Identical Polynomials

Let’s consider the two 5x5 matrices A and B provided in the example:

Matrix A

[A begin{pmatrix} 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 end{pmatrix}]

Matrix B

[B begin{pmatrix} 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 end{pmatrix}]

Both matrices have the same characteristic polynomial and minimal polynomial:

[plambda lambda^5 - 1][mlambda lambda^2 - 1]

Despite the presence of the same polynomials, the matrices are not similar because of the difference in their Jordan forms. Matrix A has a single Jordan block of size 5 for the eigenvalue 1, while matrix B has a Jordan block of size 4 and a separate block of size 1 for the eigenvalue 1.

Another Example

Let's consider another pair of matrices, A and B, which further illustrate this phenomenon:

Matrix A

[A begin{pmatrix} lambda 1 0 0 0 0 lambda 0 0 0 0 0 lambda 0 0 0 0 0 lambda 0 0 0 0 0 lambda end{pmatrix}]

Matrix B

[B begin{pmatrix} lambda 1 0 0 0 0 lambda 0 0 0 0 0 lambda 1 0 0 0 0 lambda 0 0 0 0 0 lambda end{pmatrix}]

Both matrices are distinct Jordan Canonical Forms. Matrix A has one (2 times 2) Jordan block, whereas matrix B has two of them. Because of this difference, they are not similar. However, both matrices share the same characteristic and minimal polynomials:

[chilambda lambda^5 - 1][mlambda lambda^2 - 1]

The characteristic polynomial of both matrices is the same because they are upper triangular and their diagonal elements are all (lambda). The minimal polynomial is the same because the size of their largest Jordan blocks is 2.

Conclusion

This exploration confirms that it is indeed possible for two 5x5 matrices to have the same characteristic and minimal polynomials but still not be similar. The subtle differences in their Jordan forms play a critical role in determining their similarity.

Frequently Asked Questions

Q: What is the characteristic polynomial?

A: The characteristic polynomial is a polynomial derived from the determinant of (A - lambda I), where (A) is the matrix, (lambda) is an eigenvalue, and (I) is the identity matrix. It provides information about the eigenvalues of the matrix.

Q: What is the minimal polynomial?

A: The minimal polynomial is the monic polynomial of the lowest degree such that (m(A) 0). It divides the characteristic polynomial and provides information about the structure of the matrix.

Q: What does it mean for matrices to be similar?

A: Two matrices (A) and (B) are similar if there exists an invertible matrix (P) such that (B P^{-1}AP). Similar matrices have the same characteristic polynomial and minimal polynomial, but the converse is not necessarily true.