Finding Eigenvectors: Fast and Nontraditional Approach

Abstract: Diagonalizing a matrix $A$, that is finding two matrices $P$ and $D$ such that $A = PDP^{-1}$ with $D$ being a diagonal matrix needs two steps: first find the eigenvalues and then find the corresponding eigenvectors. We show that we do not need the second step when diagonalizing matrices with a spectrum, $\left|\sigma(A)\right|\leq 2$ since those vectors already appear as nonzero columns of the $\textit{eigenmatrices}$, a term defined in this work. We further generalize this for matrices with $\left|\sigma(A)\right|> 2$ and show that eigenvectors lie in the column spaces of eigenmatrices of the complementary eigenvalues, an approach without using the classical Gauss-Jordan elimination of rows of a matrix. We introduce two major results, namely, the $\textit{2-Spectrum Lemma}$ and the $\textit{Eigenmatrix Theorem}$. As a conjecture, we further generalize the Jordan canonical forms for a new class of generalized eigenvectors that are produced by repeated multiples of certain eigenmatrices. We also provide several shortcut formulas to find eigenvectors that does not use echelon forms. The method discussed in this work may be summarized with the mnemonic "Find your puppy at your neighbors'!" argument, where puppy is the eigenvector and the neighbors are the complementary eigenmatrices.

• The paper presents a nontraditional method finding eigenvectors via eigenmatrices' column spaces, avoiding row reduction.
• The method improves computational efficiency and reduces complexity by avoiding computationally expensive row operations.
• This method can simplify teaching eigenvector calculation and offers potential for future research, including non-diagonalizable matrices.

Fast and Nontraditional Methods for Finding Eigenvectors

This paper presents a novel methodology for finding eigenvectors of diagonalizable matrices, deviating from traditional row-reduction techniques typically employed in linear algebra. The approach hinges on the concept of "eigenmatrices," a novel term introduced by the author, Udita N. Katugampola. Instead of relying on Gaussian elimination to derive eigenvectors from a basis of the null space of the matrix $A - \lambda I$, this work posits that eigenvectors can be found directly within the column spaces of certain matrices associated with complementary eigenvalues. This provides a computationally efficient alternative to conventional methods.

The primary focus lies in diagonalizable matrices with a spectrum size $j\lambda(A)j = 2$. The paper stipulates that the necessary eigenvectors naturally appear as non-zero columns in eigenmatrices of the complementary eigenvalues, obviating the need for row operations. For matrices where the number of eigenvalues $j\lambda(A)j > 2$, the study generalizes the method by utilizing the column spaces of these eigenmatrices without employing classical Gaussian elimination.

The research highlights several practical advantages of this nontraditional method:

1. Efficiency in Computation: The method reduces the need to convert matrices into their echelon forms, which can be computationally expensive, especially for larger matrices. By identifying eigenvectors from eigenmatrices, the process becomes streamlined.
2. Reduction in Computational Complexity: The method alleviates the computational burden by sidestepping complex row operations inherent in traditional methods. Instead, eigenvectors are computed through simple inspection of column spaces, significantly reducing the computational complexity.
3. Simplification Through Mnemonics: An anecdotal mnemonic device, "Find your puppy at your neighbors’!", metaphorically encapsulates the approach, emphasizing that eigenvectors of a given eigenvalue are located within the column space of the corresponding eigenmatrix.

Moreover, the paper acknowledges historical contributions to the development of spectral theory, tracing the evolution of the terms eigenvalue and eigenvector from their initial conceptualization to their current mathematical framework. The author recognized significant contributions from historical figures such as Daniel Bernoulli and James J. Sylvester for laying the groundwork for modern linear algebra.

A noteworthy feature of this research is the emphasis on computational pedagogy, proposing that this fast and intuitive methodology could be incorporated into classroom settings to facilitate understanding and provide empirical computational tools for students. The paper suggests that this approach may hold particular value when used to teach eigenvector calculation in courses such as linear algebra and differential equations, particularly in settings where computational resources or time may be limited.

In addition, the research opens potential avenues for further study, postulating on preliminary findings related to non-diagonalizable matrices. The method hints at revealing generalized eigenvectors, providing a bridge to potential expansion into the study of Jordan canonical forms and the analysis of more complex linear algebraic structures.

The paper proposes a philosophical reflection on traditional methods, prompting questions about the entrenched reliance on them despite evident computational inefficiencies. This reflective stance encapsulates the novel approach's primary intention: to challenge existing methodologies by offering a practical and effective alternative.

In conclusion, this work makes a compelling case for reevaluating conventional methodologies in spectral theory and eigenvector calculation. By presenting a computationally efficient and conceptually simpler method, it invites both scholars and educators to reconsider and potentially reform established practices in linear algebra pedagogy and computational applications. While promising, these methods stimulate speculation and discussion on their broader implications, inviting future research to validate and extend these findings within a broader mathematical and practical context.