Diagonalizability of Square Matrices: Insights into Eigenvalues and Their Multiplicities

Eigenvalues play a crucial role in the analysis and understanding of linear transformations and square matrices. Among the many properties of eigenvalues, one of the fundamental findings is the relationship between diagonalizability and eigenvalues. Specifically, if a square matrix is diagonalizable, then each eigenvalue has a geometric multiplicity that matches its algebraic multiplicity. This article delves into the implications of these multiplicities and how they relate to the minimal polynomial of the matrix.

Introduction to Eigenvalues and Diagonalizability

A square matrix A can be diagonalized if it is similar to a diagonal matrix. This means there exists an invertible matrix P such that

[ A PDP^{-1}, ]

where D is a diagonal matrix. In this context, the diagonal entries of D are the eigenvalues of A. The process of diagonalization allows us to simplify complex matrix operations and understand the inherent structure of A more clearly.

Geometric Multiplicity and Algebraic Multiplicity

The algebraic multiplicity of an eigenvalue λ is the number of times λ is a root of the characteristic polynomial of A. On the other hand, the geometric multiplicity of λ is the dimension of the eigenspace associated with λ, which is the null space of the matrix (A - λI). Mathematically, the geometric multiplicity of λ is given by:

[ text{Geometric multiplicity of } λ dim(text{null } (A - λI)). ]

A key insight is that for a square matrix to be diagonalizable, its geometric multiplicity must equal its algebraic multiplicity for every eigenvalue. This equivalence is a fundamental property of diagonalizability.

Algebraic and Geometric Multiplicities: Properties and Implications

When the algebraic multiplicity and geometric multiplicity of an eigenvalue align, it signifies that the matrix has a complete set of linearly independent eigenvectors. Let's explore some implications of this property.

Implication for Diagonalizability

Suppose a square matrix A has distinct eigenvalues or eigenvalues with algebraic and geometric multiplicities that match. Then A can be diagonalized. This is because for each eigenvalue λ, the eigenspace associated with λ has a dimension that matches its algebraic multiplicity, ensuring a sufficient number of linearly independent eigenvectors. These eigenvectors can then form the columns of the matrix P, allowing the diagonalization process to proceed.

The Minimal Polynomial

The minimal polynomial of a matrix is the monic polynomial of least degree that annihilates the matrix, i.e., it is the polynomial of lowest degree Q(x) such that Q(A) 0. One important property of the minimal polynomial is that it is closely related to the algebraic multiplicity of the eigenvalues. If a square matrix A is diagonalizable, then its minimal polynomial has no repeated roots. The roots of the minimal polynomial correspond to the eigenvalues of A, and the multiplicity of each root in the minimal polynomial is the size of the largest Jordan block associated with that eigenvalue in the Jordan canonical form. Since a diagonal matrix has Jordan blocks of size 1 for each eigenvalue, the minimal polynomial of a diagonalizable matrix will have each eigenvalue with multiplicity one.

Examples and Applications

Consider a simple example where the matrix A is:

[ A begin{pmatrix} 3 0 0 3 end{pmatrix}. ]

This matrix is already diagonal, and its eigenvalues are 3 with algebraic multiplicity 2. The geometric multiplicity of 3 is also 2, as the eigenspace associated with 3 is spanned by the standard basis vectors of (mathbb{R}^2).

For a more complex example, consider the matrix:

[ B begin{pmatrix} 4 1 0 0 4 0 0 0 1 end{pmatrix}. ]

The eigenvalues are 4 (with algebraic multiplicity 2) and 1 (with algebraic multiplicity 1). The geometric multiplicity of 4 is 2, as the eigenspace associated with 4 is spanned by two eigenvectors, and the geometric multiplicity of 1 is 1.

Conclusion

The relationship between diagonalizability and the multiplicities of eigenvalues is a cornerstone of linear algebra. Specifically, for a square matrix to be diagonalizable, the geometric multiplicity of each eigenvalue must match its algebraic multiplicity. This implies that the minimal polynomial of the matrix has no repeated roots and that the matrix can be decomposed into a diagonal form using a similarity transformation. Such understanding is not only fundamental but also highly useful in the analysis of linear systems and transformations in various scientific and engineering applications.