[Paper Review] A Tutorial on Matrix Perturbation Theory (using compact matrix notation)

Bassam Bamieh|arXiv (Cornell University)|Feb 11, 2020

Matrix Theory and Algorithms2 references4 citations

TL;DR

This paper presents a matrix notation-based tutorial on analytic perturbation theory for eigenvalues and eigenvectors of matrices, reformulating standard perturbation expansions using compact matrix equations—particularly Sylvester equations—enabling unified, simplified treatment of non-Hermitian, degenerate, and higher-order cases. The key contribution is a systematic matrix formulation that streamlines derivations and provides explicit, compact expressions for perturbation terms via solvability conditions and structured matrix solutions.

ABSTRACT

Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In this note, we give an introduction to this method that is independent of any physics notions, and relies purely on concepts from linear algebra. An additional feature of this presentation is that matrix notation and methods are used throughout. In particular, we formulate the equations for each term of the analytic expansions of eigenvalues and eigenvectors as {\em matrix equations}, namely Sylvester equations in particular. Solvability conditions and explicit expressions for solutions of such matrix equations are given, and expressions for each term in the analytic expansions are given in terms of those solutions. This unified treatment simplifies somewhat the complex notation that is commonly seen in the literature, and in particular, provides relatively compact expressions for the non-Hermitian and degenerate cases, as well as for higher order terms.

Motivation & Objective

• To provide a self-contained, linear algebra-based introduction to matrix perturbation theory independent of physics or quantum mechanics.
• To simplify and unify the treatment of eigenvalue and eigenvector perturbations using compact matrix notation.
• To derive explicit, compact expressions for higher-order perturbation terms in non-Hermitian and degenerate cases.
• To reformulate perturbation expansions as matrix equations—specifically Sylvester equations—enabling systematic solution techniques.
• To demonstrate that matrix-based formulations yield clearer insight and more concise expressions than traditional vector-by-vector approaches.

Proposed method

• Formulate eigenvalue and eigenvector relations as matrix equations: $AV = V ilde{\Lambda}$ and $W^*A = \tilde{\Lambda}W^*$, where $V$ and $W^*$ are matrices of right and left eigenvectors, and $\tilde{\Lambda}$ is the diagonal matrix of eigenvalues.
• Use matrix notation to express perturbation expansions of eigenvalues and eigenvectors as power series in $\epsilon$, leading to matrix equations for each order of $\epsilon$.
• Model each perturbation term as a solution to a Sylvester matrix equation, with solvability conditions derived from the structure of the eigenvalue spectrum.
• Apply the pseudo-inverse of the matrix $\mathbf{\Pi}^\dagger$ with entries $1/(\lambda_{0i} - \lambda_{0j})$ for $i \neq j$ and zero otherwise, to solve for eigenvector corrections.
• Use the Hadamard product and diagonal projection operators to express matrix products and extract components, particularly in degenerate cases.
• Handle degenerate eigenvalues by partitioning the matrix into blocks and ensuring $\Lambda_1$ remains diagonal through proper choice of $V_0$.

Experimental results

Research questions

• RQ1 How can matrix notation simplify the derivation and expression of perturbation terms for eigenvalues and eigenvectors?
• RQ2 What is the role of Sylvester equations in formulating higher-order perturbation expansions?
• RQ3 How can the degenerate eigenvalue case be treated uniformly using matrix formulations?
• RQ4 What conditions ensure the solvability of perturbation equations in non-Hermitian and degenerate settings?
• RQ5 How does the use of compact matrix notation reduce complexity compared to traditional component-wise derivations?

Key findings

• The paper derives a compact matrix formulation for the first-order eigenvector correction as $V_1 = -V_0(\mathbf{\Pi}^{\dagger\circ} \circ (W_0^*A_1V_0))$, where $\mathbf{\Pi}^{\dagger\circ}$ is the entrywise inverse of eigenvalue differences.
• For degenerate eigenvalues, the condition that $\Lambda_1$ must be diagonal forces $V_0$ to be chosen as the eigenvector matrix of the block $(A_1)_{11}$, ensuring consistency of the perturbation series.
• The method yields explicit, closed-form expressions for perturbation terms in both non-Hermitian and degenerate cases, avoiding the complexity of component-wise summations.
• The use of matrix notation and the Hadamard product allows for a unified treatment of all cases, including higher-order terms, with significantly reduced notational overhead.
• The solvability condition for the Sylvester equation is naturally embedded in the structure of $\mathbf{\Pi}^{\dagger\circ}$, which vanishes on diagonal entries when eigenvalues are equal.
• The approach provides a systematic framework to compute perturbation terms order-by-order, with each term expressed as a solution to a matrix equation with known solvability criteria.