Introduction to Eigenvectors of Non-Diagonalizable Matrices

Understanding how to find eigenvectors of a square matrix when it's not diagonalizable can be a challenging task. Despite the extra complexity, the process is remarkably similar to that of a diagonalizable matrix. However, in non-dia-
onalizable cases, one must carefully account for the difference between algebraic and geometric multiplicities. This article will guide you through finding eigenvectors for non-diagonalizable matrices through a step-by-step process and introduction to the Jordan Normal Form method.

Understanding the Basics

To fully comprehend the steps involved, let's begin with a brief overview of eigenvalues, eigenvectors, and the properties of diagonalizable and non-diagonalizable matrices.

Diagonalizability and Its Implications

A square matrix is diagonalizable if it can be transformed into a diagonal matrix (one with all off-diagonal elements equal to zero) through similarity transformation. This transformation is expressed as $A PDP^{-1}$, where $A$ is the original matrix, $D$ is the diagonal matrix of eigenvalues, and $P$ is the matrix of eigenvectors of $A$. For a $n$-dimensional matrix to be diagonalizable, it must have $n$ linearly independent eigenvectors. However, for some matrices, especially those that are not full-rank, the number of linearly independent eigenvectors is less than $n$, leading to non-diagonalizability.

Characteristics of Non-Diagonalizable Matrices

In non-diagonalizable matrices, at least one eigenvalue will have an algebraic multiplicity greater than its geometric multiplicity. The algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the eigenspace corresponding to that eigenvalue. For a matrix to be diagonalizable, these two multiplicities must be equal for all eigenvalues.

Step-by-Step Process for Finding Eigenvectors

Given a non-diagonalizable matrix $A$, here are the steps to find its eigenvectors:

Find the Characteristic Polynomial: Start by finding the eigenvalues of the matrix. This involves solving the characteristic equation $|A - lambda I| 0$, where $lambda$ represents the eigenvalues and $I$ the identity matrix. Verify Algebraic and Geometric Multiplicities: For each eigenvalue, compute the algebraic multiplicity by counting how many times the eigenvalue appears in the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace, which is the nullity of the matrix $(A - lambda I)$. Find the Eigenspaces: Solve the equation $(A - lambda I) mathbf{x} mathbf{0}$ to find the eigenvectors corresponding to each eigenvalue. The solutions to this equation form the eigenspace. Complete the Basis: If the geometric multiplicity of an eigenvalue is less than its algebraic multiplicity, find generalized eigenvectors to complete the basis of the eigenspace. This is particularly important for eigenvalues with algebraic multiplicity greater than 1. Construct the Jordan Normal Form: Once all eigenvectors and generalized eigenvectors are found, you can construct the Jordan Normal Form of the matrix if needed for further analysis.

The Jordan Normal Form Method

The Jordan Normal Form (JNF) is a particularly useful method for dealing with non-diagonalizable matrices. The JNF of a matrix is a block diagonal matrix, where each block corresponds to an eigenvalue. Each block is a Jordan block, which has the eigenvalue on the diagonal and ones on the superdiagonals (the diagonals just above the main diagonal). Here’s why the JNF is important:

Dimension: Each Jordan block has a size corresponding to the algebraic multiplicity of the eigenvalue. The sum of these sizes gives the total dimension of the eigenspace. Eigenvectors and Generalized Eigenvectors: The Jordan blocks allow for a clear distinction between the eigenvectors and the generalized eigenvectors. For an eigenvalue with algebraic multiplicity $m$, the JNF has $m$ Jordan blocks. Constructing the Matrix: The transition matrix $P$ from the matrix $A$ to its JNF $J$ is the matrix whose columns are the eigenvectors and generalized eigenvectors of $A$. The JNF $J$ can then be expressed as $A PJP^{-1}$.

Practical Examples and Applications

Understanding the method through practical examples is crucial. Let's walk through an example to solidify the concepts:

Example: Finding Eigenvectors and JNF of a Non-Diagonalizable Matrix

Consider the matrix $A begin{bmatrix} 1 1 0 1 end{bmatrix}$. The characteristic polynomial is obtained by solving $|A - lambda I| 0$, which gives $(1-lambda)^2 0$. Therefore, the only eigenvalue is $lambda 1$ with algebraic multiplicity 2.

Solve $(A - lambda I) mathbf{x} mathbf{0}$ for $lambda 1$ to find the geometric multiplicity. We get $begin{bmatrix} 0 1 0 0 end{bmatrix} begin{bmatrix} x_1 x_2 end{bmatrix} begin{bmatrix} 0 0 end{bmatrix}$, which has infinitely many solutions given by $x_1 0$. Thus, the geometric multiplicity is 1 and the eigenspace is spanned by $begin{bmatrix} 0 1 end{bmatrix}$. To find the complete set of eigenvectors, we need one more vector. We solve $(A - lambda I) mathbf{v} mathbf{x}$ to find a generalized eigenvector $mathbf{v}$ such that $(A - lambda I) mathbf{v} mathbf{x}$. Setting $mathbf{x} begin{bmatrix} 1 0 end{bmatrix}$, we solve $begin{bmatrix} 0 1 0 0 end{bmatrix} begin{bmatrix} v_1 v_2 end{bmatrix} begin{bmatrix} 1 0 end{bmatrix}$, which gives $v_2 1$ and $v_1$ free. Choosing $v_1 0$, we get $mathbf{v} begin{bmatrix} 0 1 end{bmatrix} t begin{bmatrix} 1 0 end{bmatrix} begin{bmatrix} 1 0 end{bmatrix}$ (for $t 1$). Form the transition matrix $P begin{bmatrix} 0 1 1 0 end{bmatrix}$, and compute the JNF $J begin{bmatrix} 1 1 0 1 end{bmatrix}$. Then, $A PJP^{-1}$.

Conclusion

The process of finding eigenvectors of non-diagonalizable matrices involves careful consideration of algebraic and geometric multiplicities and the use of the Jordan Normal Form to find a complete set of eigenvectors and generalized eigenvectors. The Jordan Normal Form is a powerful tool that allows for a deeper understanding of matrix properties and is widely used in various fields, including physics, engineering, and computer science.

For further study, you may want to explore additional resources such as textbooks on linear algebra, online courses, and research papers. Understanding the intricacies of non-diagonalizable matrices and their applications is crucial for anyone dealing with complex mathematical systems.